Accessibility of the lower-hybrid resonance

Transcription

1 Accessibility of the lower-hybrid resonance [Sec. 4.1, Stix] Mathematica initialization Basic definitions Ion-to-electron mass ratio: Μ = All frequencies are assumed measured in units of Ωpe, so: Ωpi := Μ-1 Ωpe Ratio of the electron cyclotron frequency (absolute value) to the electron plasma frequency: We := - Μ Wi Ωpe L@Ω_D := 1 - Ω HΩ - We L Ωpi - Ωpe R@Ω_D := 1 1 S@Ω_D := D@Ω_D := 1 Ω HΩ + We L Ω HΩ - Wi L Ωpi - Ω HΩ + Wi L HR@ΩD + L@ΩDL HR@ΩD - L@ΩDL Ωpe P@Ω_D := 1 - Ω Accessibility of the lower-hybrid range at kèè = Let us plot the refraction index, n, for the X wave in the lower-hybrid frequency range. n@ω_d := R@ΩD L@ΩD S@ΩD

2 Plot3DA E n@ωd. 9Ωpe x, Wi y Μ, Ω 1=, 8x,.1, 15<, 8y,.1, <, PlotPoints 1, ImageSize 4, Mesh None, Filling Bottom, AxesStyle 16, AxesLabel 9"Ωpe Ω", " We Ω", "n"= Same in the form reminiscent of the CMA diagram:

3 3 DensityPlotA E n@ωd. 9Ωpe x, Wi y Μ, Ω 1=, 8x,.1, 15<, 8y,.1, <, PlotPoints 15, ImageSize 3, Mesh None, FrameLabel 9"Ωpe Ω", " We Ω", "n"= In principle, one may be able, as in the case of EBW heating, to start out with the high-b side. That would require starting with Ω ~ Wi out and ending up with Ω ~ ΩLH ~ Wi out ~ Wi We. Since Ω is fixed, this requires Wi We, or, equivalently, Wi out ~ Wi We ~ mi p 1. me Wi Actual tokamaks do not enjoy such a large variation of the magnetic field strength across plasma. Thus, as long as we stick to zero kèè, we have to tunnel the energy across a major frequency gap. That is problematic. Accessibility of the lower-hybrid range at kèè ¹ Code a@ω_d := S@ΩD b@ω_d := R@ΩD L@ΩD + P@ΩD S@ΩD - P@ΩD q - S@ΩD q c@ω_d := P@ΩD IR@ΩD L@ΩD - S@ΩD q + q M := b@ωd + b@ωd - 4 a@ωd c@ωd a@ωd

4 := a@ωd c@ωd a@ωd Μ = Pic@Q_, R_D := BlockA9Wi = 1.1 Μ-1, Ω = 1, q = Q, uh=, uh = AbsAx. NSolveAS@1D. 9Ωpe x=, xee@@1dd Ωpe x Ω, Ωpe x Ω=, 8x,.1, 4<, PlotStyle Directive@BlueD, AspectRatio 1, PlotRange R, ImageSize 5, GridLines 88, uh<, 8<<, GridLinesStyle Directive@Dashed, RedD, PlotPoints, RegionFunction Function@8x<, Abs@x - uhd >.1D, FrameLabel 9"Ωpe Ω", " "= EE Illustrations A much easier way is to use nonzero kèè. For this case, the theory is described in Sec. 4.1 in [Stix]. Below, we offer only some illustrations. The following figures show versus Ωpe (i.e., essentially, versus the plasma density) for three different values of nèè. (The dashed line show the zeros and the location of the lower-hybrid resonance.) GraphicsRow@8Pic@, 8-5, 6<D, Pic@5.7, 8-3, 1<D, Pic@5.8, 8-3, <D<D Ωpe Ω Ωpe Ω 3 4 It is seen that there exists a critical nèè at which the upper branch, which approaches the lower-hybrid resonance, extends continuously to low densities. However, it is also seen that this branch does not extend to zero density but rather has a low-density cutoff at Ω = Ωpe. (In this case, P =, so c =, so =. Since the wave still has a nonvanishing nèè, it represents a Langmuir wave near this cutoff.) That does represent a problem for launching this wave from outside the plasma. Yet, in tokamaks, the small-density region where the wave cannot propagate is relatively small (~ cm), and such a small gap is not impossible to deal with. Comment 1: Note that branches that would correspond to the separate O and X waves are now connected. How is this possible? Well, remember that the dispersion relation is Ω = ΩHkÞ, kèè L, which determines a surface in the 3D k-space. Dispersion curves are obtained as sections of this surface by surfaces corresponding to fixed Θ or fixed kèè. The topology of those curves can vary depending on which section of the surface is considered i.e., what looks like disconnected branches in one section looks like connected branches in another section. In

5 5 Comment 1: Note that branches that would correspond to the separate O and X waves are now connected. How is this possible? Well, remember that the dispersion relation is Ω = ΩHkÞ, kèè L, which determines a surface in the 3D k-space. Dispersion curves are obtained as sections of this surface by surfaces corresponding to fixed Θ or fixed kèè. The topology of those curves can vary depending on which section of the surface is considered i.e., what looks like disconnected branches in one section looks like connected branches in another section. In particular, X and O waves are, in fact, connected and thus can be transformed into each other adiabatically. Comment : Even though the model of fixed nèè is widely used in literature, in application to LHCD it is generally incorrect. Conserved is actually the toroidal component of the refraction index, nφ, which is not the same as nèè. Since LHCD is usually done at p nèè, even a small angle between the toroidal direction and the direction along the static magnetic field causes a major change in nèè. For some details, see the following figure [from X. Guan, I. Y. Dodin, H. Qin, J. Liu, and N. J. Fisch, On plasma rotation induced by waves in tokamaks, Phys. Plasmas, 115 (13)]: