P.B. ALEYNIKOV and N.B. MARUSHCHENKO

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ECRH AND MODE CONVERSION IN OVERDENSE W7-X PLASMAS P.B. ALEYNIKOV Max-Planck-Institut für Plasmaphysik Greifswald, Germany Email: pavel.aleynikov@ipp.mpg.de

N.B. MARUSHCHENKO Max-Planck-Institut für Plasmaphysik Greifswald, Germany

Abstract

The wave physics of O- to X-mode conversion in over-dense Wendelstein7-X plasmas is investigated using 3D full-wave calculations. The feasibility of mode conversion scenarios within the capabilities of the existing ECRH system in Wendelstein 7-X is considered. An optimisation of the O- to X- conversion efficiency within the constraints set by the 3D plasma equilibrium is demonstrated. A heating scenario with >85% efficiency is identified. The effect of turbulence on the conversion efficiency is explored and is found to be relatively small for the Wendelstein 7-X-relevant turbulence parameters. The results are also applicable to the inverse problem of electron Bernstein emission (EBE) diagnostics.

1. INTRODUCTION

Electron Cyclotron Resonance Heating (ECRH) is the main plasma heating mechanism in the Wendelstein 7-X (W7-X) Stellarator. It is provided by 10 gyrotrons at 140 GHz (corresponding to the second harmonic cyclotron resonance at 2.5 T) with the power of ~1 MW each. X2- and O2-modes were successfully used in a wide range of operation scenarios: X2-mode for low and moderate densities (up to the cutoff at 1.2×1020 m-3), and O2-mode for higher densities (up to ~2×1020 m-3). A schematic diagram illustrating O- and X-mode cutoffs and corresponding heating scenarios is shown in Fig. 1.

Possible operation at yet higher densities would involve double mode-conversion from O- to slow X- and then to a Bernstein-mode [1], i.e. an OXB-scenario (indicated schematically with green arrows in Fig. 1). The physics of O-X conversion is outside of applicability of the routinely used geometrical optics approximation (WKB-theory) and should be considered within a full-wave approach [2,3].

FIG. 1. Schematic diagram of the O- and X-modes cutoffs (black and blue dashed and solid curves) and corresponding heating scenarios (thick red, green and blue arrows).

The efficiency of the entire OXB scheme is defined solely by the efficiency of the O-X conversion. Therefore, in this work, we focus on the wave physics of O-X conversion. For this study, a new 3D cold plasma full-wave code named “CUWA” has been developed. The code utilizes the Finite Difference Time Domain (FDTD) technique to solve the initial-value problem for Maxwell’s equations coupled with the cold plasma response

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[4,5]. The obtained solution is, therefore, a time-dependent distribution of the wave-fields in three dimensions. If required, the wave properties (such as the wave vector) can be derived from the solution via a Fourier transform.

The computation domain is “minimized” around the WKB-trajectory of the reference ray, and is matched to the surrounding plasma by using the so-called “convolutional perfectly matched layers” (CPML) boundary condition [6,7]. The background magnetic field is recovered from pre-computed 3D equilibrium data. The code takes advantage of massive parallel computations with Graphics Processing Units (GPUs), which allows for up to 100 times faster calculations than on a single-CPU. This feature allows for efficient parametric optimization studies over a broad range of possible experimental conditions in 3D calculations. The code has an interface with the ray-tracing code TRAVIS [8].

2. MODE CONVERSION

Efficient conversion occurs when the “turning” points for the wave-packets of the two modes (O- and slow X-) are close to each other. Formally, the conversion condition is satisfied at the intersection of two surfaces: (A) the O-mode cutoff surface (i.e. plasma frequency equal wave frequency) and (B) the X-mode turning surface parameterized by the plasma frequency, the cyclotron frequency, and the wave parallel refractive index. For each point on the O-mode cutoff surface, the WKB-theory gives exactly one optimal perfectly converted wave. However, when the effects of beam shaping and 3D curvature of the plasma surfaces (i.e. non-WKB factors) are taken into account, the conversion efficiency drops significantly.

Figure 2 shows an example of a full-wave 2D calculation of the O-X conversion process in a realistic W7-X plasma for the “WKB-optimal” ray. The O-mode, launched from vacuum (bottom-left corner of the left plot), propagates into the plasma and is partly reflected off the corresponding cut-off layer (red dashed line), where a fraction of the O-mode is converted into a slow X-mode, which propagates toward the upper-hybrid resonance (black dashed line). Eventually, this X-mode is converted to electron Bernstein wave, however, this is not shown in the figure. The wavelength of the Bernstein modes tends to be very small and such waves are numerically dissipated on the finite computation mesh (this dissipation is of purely numerical nature).

The reflected part of the O-mode has a “hole” due to the non-uniform O-X conversion. The perpendicular component of the refractive index is computed with the use of a windowed Fourier transform and is shown in Fig. 2 (right). The incoming O-mode, the reflected O-mode, and the slow-X mode (which settles near the Upper Hybrid resonance layer) can be clearly distinguished in this diagram, which appears to be in a good agreement with expectations from WKB theory (i.e. dispersion curves), validating the full-wave solution.

FIG. 2. An example of the O-X mode conversion. Left plot: the wave electric field (colour-coded) and the ray tracing calculation (solid red). The antenna is located in the left bottom corner. O-mode cutoff and Upper Hybrid resonance layers are plotted with the red and the black dashed lines, respectively. Right plot: windowed Fourier transform of the wave field,

i.e. perpendicular wave refractive index (colour-coded) and the refractive index predicted by the ray-tracing (solid).

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The W7-X plasma is strongly shaped (as is, therefore, the evanescent layer between surfaces A and B) and generally requires a 3D evaluation of the conversion process. In Fig. 3, the 3D calculation of the above scenario is shown (the wave is launched from the bottom of the domain). It is evident from the cross-section of the reflected wave (vertical cross sections) that the 3D geometry needs to be appropriately accounted for. A Gaussian beam is used as an initial condition. Yet, the reflected O-mode has a complex 2D profile. The ratio of the time-averaged fluxes of the Poynting vector through the boundaries of the computational domain in steady-state reflects the O-X conversion efficiency (T). In this calculation T ≈ 0.6.

FIG. 3. 3D calculation of the scenario described in Fig. 2. The decimal logarithm of the magnitude of the time-averaged wave Poynting vector is colour-coded. The ray-tracing trajectory is shown with the green curve. The O-mode cutoff is shown

with the black surface.

3. MODE CONVERSION IN W7-X

The ECRH antenna setup in W7-X is shown schematically in Fig. 4 (left) (orange planes); the O-mode cutoff surface is also shown with the magnetic field strength colour-coded. For the six steerable antennas (labelled, top to bottom: F1, N1, D1 in the left column and A1, C1, E1 in the right), a ray-tracing study gives 12 “WKB-optimal” rays (red in Fig. 4 (left)), two rays for each antenna, launched co- and counter to the magnetic field. Those are the rays that encounter efficient O-X conversion conditions on their trajectory. As demonstrated above, ray-tracing can only be used to determine approximate launch parameters (the azimuthal and vertical aiming angles) of such rays, because the actual 3D shape of the evanescent layer and of the beam properties cannot be taken into account within the WKB-theory.

Therefore, for a realistic prediction of the conversion efficiency achievable in W7-X, we run a large set of 3D full-wave calculations varying the aiming angles of the antenna around the those predicted by ray-tracing. Such a set of calculations is produced for each of the 12 “WKB-optimal”-rays.

The overall result of this study is shown in Fig. 4 (right plot). The plot demonstrates contours of the conversion efficiency (T) (colour-coded) for a range of aiming angles ([-2.5º; +2.5º]) around each of the 12 “WKB-optimal”-rays. The launching antenna label and the aiming angles of “WKB-optimal”-rays are indicated above each of the contour plots. We observe that, while the aiming angles predicted by the ray-tracing are relatively precise, the realistic conversion efficiency is significantly lower than 100% in all cases.

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FIG. 4. Left plot: ECRH antennas (orange) in W7-X (antennas labelled, top to bottom: F1, N1, D1 in the left column and A1, C1, E1 in the right column) and the O-mode cutoff surface. Colour-coded is the magnetic field strength. Red curves

represent rays with high O-X conversion predicted by ray-tracing (“WKB-optimal” rays). Right plot: The contours of the conversion fraction (colour-coded) for a range of aiming angles ([-2.5º +2.5º]) around each of the 12 “WKB-optimal”-rays. The launching antenna label and the aiming angles of the corresponding “WKB-optimal”-ray are indicated above each of

the contour plots.

The conversion is consistently higher for the A1, C1, N1 and D1 mirrors (reaching more than 60% for the considered parameters), and it is around 50% for the other mirrors. Such a conversion efficiency is too low for a reasonably practicable, heating scenario and in the next section we will discuss methods for improvement.

These calculations were carried out for the standard and the “high-mirror” W7-X magnetic configurations, both yielding similar results. The beam size and the focusing properties correspond (approximately) to those of the installed W7-X beams1.

The density profile usually varies significantly even during a single experiment. Therefore it was chosen rather arbitrarily in our setup. However, we found that the variation of the location of the O-mode cutoff flux surface itself does not affect our results significantly. This is due to the fact that near the bean-shaped segment in W7-X (near the ECRH launchers) the magnetic-field structure varies only slightly between the neighbouring flux surfaces. Consequently, while aiming angles change with the variation of the cutoff, the conversion efficiency, defined by the characteristics of the evanescent layer, is only slightly affected.

On the other hand, it is known that the conversion efficiency, T, depends exponentially on the plasma density length scale ( L ≡ ne /∇ne ) [9],

T ~ exp −π ωcL⎛

⎝⎜⎞⎠⎟ ,

where ω is the wave frequency and c is the speed of light. For the calculations shown in Fig. 4 we chose the density profile such that L ≈ 1/15 m at the cutoff. For different gradients, the results should be scaled accordingly.

___________________________________________________________________________ 1 The Gaussian beam radius was somewhat reduced in our calculations compared to that of the W7-X beams. This was done in order to reduce the computational domain and thus the computational costs. However, we have verified that overall results are not significantly affected by this modification.

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4. OPTIMISED CONVERSION SCENARIO IN W7-X PLASMAS

In the previous section, we have shown that 50% is a typically achievable conversion fraction for typical W7-X equilibrium, beam parameters, and launching positions. References [10,11] suggest that conversion can be improved by a careful design of the beam wave-front. In this section, we pursue an alternative optimisation strategy, in which we “optimize” the equilibrium properties while the beam structure is kept unchanged.

We take advantage of the fact that a stellarator equilibrium “offers” a wide variety of magnetic field curvatures on the O-mode cutoff flux surface, and we search for a location which would minimise the thickness of the evanescent layer, weighted over the radial beam profile. The thickness of the evanescent layer is estimated using the results of analytical theory [9], where this thickness is taken to be the distance (in wave-vector space) between the WKB solutions of the two modes. This estimate is known to agree very well with calculations in a flat 2D case [12].

The result of this calculation is demonstrated in Fig. 5 (left). The function representing the Gaussian-weighted gradient of the thickness of the evanescent layer is plotted on the cutoff surface, which is parameterized by the poloidal (θ) and the toroidal (φ) angles. A few minima of this function are marked with red dots. A different optimal value of parallel refractive index N||opt (and thus launching angles) corresponds to each minimum due to the variation of the magnetic field strength on the surface.

FIG. 5. Left: Location of the optimal conversion regions (red dots) on the cutoff surface (φ and θ are toroidal and poloidal angles). The function representing the weighted gradient of the thickness of the evanescent layer is colour-coded. Middle:

Conversion region for N||opt ≈ 0.564. Right: Optimised efficient conversion region for N||opt ≈ 0.563 (φ ≈ 3.5 and θ ≈ 190 on the left plot).

It is easy to understand the significance of these special points. We recall that the efficient conversion occurs when the “turning” points for the wave-packets of the two modes (O- and X-) are close to each other and that the conversion condition is satisfied at the intersection of two surfaces: the O-mode cutoff surface and the X-mode turning surface parameterised by the plasma frequency, the cyclotron frequency, and the wave parallel refractive index N||. For a particular equilibrium, the intersection curve (3D-shaped, in general) will vary for waves with different parallel refractive index N||opt. Fig. 5 (middle) shows this intersection for N||opt ≈ 0.564 (an arbitrary value), where the surfaces cross each other obliquely and thus the evanescent layer is narrow only close to the intersection (if the beam has a homogenous N||= N||opt structure). When N||opt corresponds to one of the optimised locations (red dots in Fig. 5 (left)), the intersection has a more complex structure (a saddle point in the example of Fig. 5 (right)), ensuring minimisation of the evanescent layer weighted over the finite width of a Gaussian beam.

The location of the conversion region (and thus N||opt) uniquely defines the O-mode ray arriving at this point

within the WKB approximation. Therefore, the beam launching location and angles can be computed with a backward ray-tracing calculation (starting from the conversion region). When the initial conditions for the beam outside of plasma are known, full-wave calculations are conducted to predict the conversion efficiency. The maximised conversion efficiency predicted with our full-wave code for a case of Fig. 5 (right) reaches a remarkable 85% for otherwise the same plasma and beam parameters as in the cases discussed in the previous

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section. This calculation is shown in Fig. 6. The detailed structure of the evanescent layer on the cutoff surface (vertical) is shown with dark-blue colours.

Note, however, that launching such an optimized wave in W7-X requires the antenna location to be chosen freely, which is not possible in practice. Yet, this optimization approach demonstrates that high conversion efficiency is theoretically achievable for a realistic stellarator equilibrium. Moreover, it could be possible to design an equilibrium for which the optimized conversion region is accessible from a given antenna location.

FIG. 6. 3D calculation of the scenario described in Fig. 5 (right). The structure of the evanescent layer on the cutoff surface (vertical) is colour-coded with dark-blue colours. Solid curves represent the ray-tracing trajectory.

5. EFFECT OF TURBULENCE ON CONVERSION EFFICIENCY

The effect of plasma turbulence on the conversion efficiency was noticed in the pioneering OXB experiments on W7-AS [13]. Their detrimental role was conjectured in that work. However, theoretical considerations presented in Ref. [14] suggest that plasma fluctuations most likely cannot be responsible for the low efficiency of OXB heating. In the present section, we address this problem by direct numerical calculation of the O-X conversion in perturbed plasma.

The plasma density fluctuations move with velocities much smaller than the speed of light and therefore can be treated as stationary perturbations when solving for the wave fields. This necessitates averaging over a large number of individual calculations (with statistically independent realizations of the density perturbations) in order to evaluate the conversion efficiency. Thanks to its speed, the GPU-capable code CUWA allows for efficient calculation of such a large number of 3D solutions.

Figure 7 shows a snapshot of the wave field (white contours) in the OX conversion scenario with a perturbed density profile. The surfaces of constant density are shown in colour. As expected, the wave field is significantly scattered by the density fluctuations.

The turbulent layer is generated using the model of Ref. [15] (the model produces random Gaussian fluctuations with given correlation length and amplitude). We impose fluctuations with a uniform relative amplitude (δne/ne) over the entire density profile.

In Fig. 8 saturation of the conversion fraction T (or, equivalently, the reflected intensity) with the growth of the number of integrated snapshots is shown. The value saturates when the number of integrated snapshots exceeds ~150. In further studies, we use averaging over 150 snapshots.

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FIG. 7. The snapshot of the wave field (white contours) in the O-X conversion scenario with perturbed density profile. The surfaces of constant density are shown in colour.

FIG. 8. Reflected O-mode intensity averaged over the growing number of snapshots.

Figure 9 shows the dependence of the conversion efficiency on the relative amplitude and the correlation length of the imposed fluctuations for the optimized scenario discussed in the previous section. In W7-X the fluctuations are expected to have the correlations length of order 1 cm and a relative amplitude not higher than 10%. These calculations show that for the W7-X conditions the effect of turbulence on the O-X conversion is quite moderate, reaching the maximum of 10% loss of efficiency at low correlation lengths and high amplitude of the fluctuations.

6. CONCLUSION

The O-X conversion process defines efficiency of the entire OXB heating scheme, which is suitable for heating of overdense plasmas. In this work, we conducted a comprehensive study of the O-X conversion efficiency in realistic W7-X parameters. For realistic W7-X equilibrium and ECRH parameters, the efficiency of the scheme is low and is not expected to exceed 50-60%. However, we also demonstrated that a significant improvement of the conversion could be achieved if the equilibrium is tailored in such a way that the conversion occurs at the location where the evanescent layer width weighted over the beam profile is minimized.

It was also shown that the density fluctuations expected in W7-X should not have a significant effect (< 10%) on the O-X conversion efficiency.

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FIG. 9. Dependence of the conversion efficiency on the amplitude and the correlation length of the plasma density fluctuations.

ACKNOWLEDGEMENTS

We would like to acknowledge useful discussions with Torsten Stange, Yuriy Turkin and Per Helander.

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