Abstract
The paraxial Wentzel–Kramers–Brillouin (pWKB) approximation, also called beam tracing method, has been employed in order to study the propagation of lower hybrid waves in a tokamak plasma. Analogous to the wellknow ray tracing method, this approach reduces Maxwell’sequations to a set of ordinary differential equations, while, in addition, retains the effects of the finite beam crosssection, and, thus, the effects of diffraction. A new code, LHBEAM (lower hybrid BEAM tracing), is presented, which solves the pWKB equations in tokamak geometry for arbitrary launching conditions and for analytic and experimental plasma equilibria. In addition, LHBEAM includes linear electron Landau damping for the evaluation of the absorbed power density and the reconstruction of the waveelectric field in both the physical and Fourier space. Illustrative LHBEAM calculations are presented along with a comparison with the ray tracing code GENRAY and the full wave solver TORICLH.
The many insightful discussions with M. Bornatici, R. Bilato, and M. Brambilla are gratefully acknowledged. This research is supported by the U.S. Department of Energy under Contract No. DEFC0299ER54512.
I. INTRODUCTION AND BACKGROUND
II. PARAXIAL WKB METHOD
III. CODE DESCRIPTION
IV. RESULTS AND BENCHMARK
A. Numerical results
B. Benchmark with GENRAY and TORICLH
V. SUMMARY AND DISCUSSION
Key Topics
 Plasma waves
 66.0
 Ray tracing
 19.0
 Electric fields
 17.0
 Geometrical optics
 15.0
 Maxwell equations
 14.0
H05H1/02
Figures
(a) 3D evolution of the LH wave beam for keV and . Attenuation ellipses (mentioned in Sec. II) are plotted (in blue), together with the beam axis (in red). (b) A zoomin of (a) from another point of view. The large (green) arrows represent the wave vector direction along the LH propagation pointing to the high field side, while the small (black) arrows represent the direction of the magnetic field opposite to the direction of the LH wave beam.
(a) 3D evolution of the LH wave beam for keV and . Attenuation ellipses (mentioned in Sec. II) are plotted (in blue), together with the beam axis (in red). (b) A zoomin of (a) from another point of view. The large (green) arrows represent the wave vector direction along the LH propagation pointing to the high field side, while the small (black) arrows represent the direction of the magnetic field opposite to the direction of the LH wave beam.
Comparison between pWKB and ray tracing methods for the LH wave propagation in the poloidal crosssection. The colored area represents the projections of the attenuation ellipses on the poloidal section obtained from LHBEAM and the solid (red) curve is the beam axis, whereas the dashed (black) curves represent 10 rays generated by GENRAY. The solid (blue) diamond and the solid (magenta) circle indicate the position along the wave beam in which the power is fully absorbed, respectively, for keV and 10 keV.
Comparison between pWKB and ray tracing methods for the LH wave propagation in the poloidal crosssection. The colored area represents the projections of the attenuation ellipses on the poloidal section obtained from LHBEAM and the solid (red) curve is the beam axis, whereas the dashed (black) curves represent 10 rays generated by GENRAY. The solid (blue) diamond and the solid (magenta) circle indicate the position along the wave beam in which the power is fully absorbed, respectively, for keV and 10 keV.
Magnitude of the parallel component of the complex electric field normalized to its value at the launching point of the reference ray for three different cases: 3 keV (a), 5 keV (b), 10 keV (c), for the fixed toroidal mode number , which corresponds to the dominant component of the spectrum, showing in (d).
Magnitude of the parallel component of the complex electric field normalized to its value at the launching point of the reference ray for three different cases: 3 keV (a), 5 keV (b), 10 keV (c), for the fixed toroidal mode number , which corresponds to the dominant component of the spectrum, showing in (d).
Poloidal (a) and toroidal (b) projection of the trajectory of the LH beam axis calculated by LHBEAM (full red curve) and the trajectory of a single ray calculated by GENRAY^{32} (dashed blue curve). Circular crosssection equilibrium, parabolic plasma profiles and the initial value of , and electron temperature keV are assumed.
Poloidal (a) and toroidal (b) projection of the trajectory of the LH beam axis calculated by LHBEAM (full red curve) and the trajectory of a single ray calculated by GENRAY^{32} (dashed blue curve). Circular crosssection equilibrium, parabolic plasma profiles and the initial value of , and electron temperature keV are assumed.
Parallel (a) and perpendicular (b) component of the refractive index as a function of R, corresponding to the case of Figure 4, calculated by LHBEAM (full red curve) and GENRAY^{32} (dashed blue curve).
Parallel (a) and perpendicular (b) component of the refractive index as a function of R, corresponding to the case of Figure 4, calculated by LHBEAM (full red curve) and GENRAY^{32} (dashed blue curve).
Poloidal projection of the trajectory of the LH beam axis calculated by LHBEAM (full red curve) and the trajectory of a single ray calculated by GENRAY^{32} (dashed blue curve). Alcator CMod like equilibrium and the initial value of are assumed.
Poloidal projection of the trajectory of the LH beam axis calculated by LHBEAM (full red curve) and the trajectory of a single ray calculated by GENRAY^{32} (dashed blue curve). Alcator CMod like equilibrium and the initial value of are assumed.
Parallel (a) and perpendicular (b) component of the refractive index as a function of R, corresponding to the case of Figure 6, calculated by LHBEAM (full red curve) and GENRAY^{32} (dashed blue curve).
Parallel (a) and perpendicular (b) component of the refractive index as a function of R, corresponding to the case of Figure 6, calculated by LHBEAM (full red curve) and GENRAY^{32} (dashed blue curve).
Power absorption profile as a function of the square root of the normalized poloidal flux, as calculated by singlepass LHBEAM (solid lines), GENRAY (dasheddotted lines), and TORICLH (dashed lines) for 3 keV (red curves), 5 keV (green curves), and 10 keV (black curves). Input power is 1 MW. Note that LHBEAM takes into account the full toroidal spectrum.
Power absorption profile as a function of the square root of the normalized poloidal flux, as calculated by singlepass LHBEAM (solid lines), GENRAY (dasheddotted lines), and TORICLH (dashed lines) for 3 keV (red curves), 5 keV (green curves), and 10 keV (black curves). Input power is 1 MW. Note that LHBEAM takes into account the full toroidal spectrum.
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