The wave energy flux of high frequency diffracting beams in complex geometrical optics

Abstract

The description of high-frequency beams in inhomogeneous dispersive media is usually dealt with by asymptotic methods that greatly simplify the computational problem. This is the case of electron cyclotron heating and current drive applications in large magnetic confinement devices, where the wave-length to plasma-size ratio is extremely small, thus, hampering the direct numerical solution of the relevant wave equation. One such method is complex geometrical optics in its eikonal-based formulation. The theoretical basis of this method, however, appears to have received relatively less consideration, so that many important aspects, such as wave energy transport, have yet to be rigorously clarified. In this work, we attempt to fill up this gap. A precise asymptotic expansion exploiting the paraxial character of complex eikonal waves is proposed, which eventually leads to an equation for the description of the wave energy transport, including the effects of diffraction. The eikonal-based formulation of complex geometrical optics provides the theoretical basis of the GRAY code [Farina, Fusion Sci. Technol. 52, 154 (2007)], which is extensively used in fusion applications. A few numerical experiments are presented, showing, in particular, that extended rays determined by GRAY accurately represent the wave energy flow, and that the same information can also be retrieved from the beam tracing code TORBEAM [Poli at al., Comp. Phys. Comm. 136, 90 (2001)].

Assessment of the ITER electron cyclotron upper launcher capabilities in view of an optimized design

“…The propagation of EC waves in ITER in the form of Gaussian beams, like those foreseen to be injected from the UL, involves quite different scale lengths: the plasma minor radius a = 2 m which is the typical length over which the plasma properties change, the beam size w designed to be of the order of a few centimeters and the wavelength λ ≃ 1.8 mm for a frequency f = 170 GHz. A clear ordering λ ≪ w ≪ a is thus always maintained, also in the plasma where the typical density n e ∼ 10 20 m −3 is much lower than the cut-off density (n e,co = 3.58 10 20 m −3 for O-mode) and asymptotic techniques like beam tracing or complex geometrical optics can be used to model the wave propagation [7,8].…”

“…A clear ordering λ ≪ w ≪ a is thus always maintained, also in the plasma where the typical density n e ∼ 10 20 m −3 is much lower than the cut-off density (n e,co = 3.58 10 20 m −3 for O-mode), and asymptotic techniques like beam tracing or complex geometrical optics can be used to model the wave propagation [7,8].…”

“…Radial location of q = 2 (----, red) and q = 3/2 (· · · · · ·, blue) are shown as reference. Case 1 scenario at t = 520 s, scaled according to (8). Only cases with more than 95% absorbed power are represented.…”

TORBEAM 2.0, a paraxial beam tracing code for electron-cyclotron beams in fusion plasmas for extended physics applications

“…The wave input is provided to RELAX in terms of information (position, parallel refractive index, power fraction and cyclotron frequency) calculated on individual rays, which in the frame of geometrical optics represent the wave energy flow. To provide an analogous information in the frame of the pWKB approach, extended rays describing the energy flow of a diffracting beam can be derived [26,39]. As known from standard geometric optics [40], the action function (phase) (integral of ds = N • dr along the rays) solves the Hamilton-Jacobi equation H(r, ∇s) = 0 and the velocity field V defined above (tangent to the rays of geometric optics) determines the wave energy transport through Eq.…”

“…(13) on each ray. When the complex-eikonal ansatz (3) is introduced, assuming λ/W ≪ 1, the relevant Hamilton-Jacobi equation becomes [41,36] H(r, ∇s) − 1 2 As far as the energy flux is concerned, it can be shown [39] that Eq. (13) still holds in the complex-eikonal case, but the function s is now a solution of the modified Hamilton-Jacobi equation (15).…”