Resonance Cone Interaction With a Self-Consistent Radio-Frequency Sheath

We study the propagation of lower-hybrid-type resonance cones in a tenuous magnetized plasma, and, in particular, their interaction with, and reflection from, the plasma sheath near a conducting wall. The sheath is modeled as a vacuum gap whose width is given by the Child-Langmuir law. The application of interest is when the resonance cones are launched (parasitically) by an ion-cyclotron radio-frequency antenna in a typical rf-heated tokamak fusion experiment. We calculate the fraction of launched voltage in the resonance cones that is transmitted to the sheath, and show that it has a sensitive thresholdlike turn on when a critical parameter reaches order unity. Above threshold, the fractional voltage transmitted to the sheath is order unity, leading to strong and potentially deleterious rf-wall interactions in tokamak rf heating experiments. Below threshold, these interactions can be avoided.

Figure 1

Resonance cone propagation from a localized source at $x=-3$, $z=1$ in the plane $z/L$ and $x/{\delta }_e$ with walls at $z=0,2$. Inset shows the launch region. Energy propagates to increasing $x$. Dotted region is the domain of the analytical calculation.

Figure 2

Top: Resonance cone interaction with a sheath at $x=z=0$ for $\Lambda =3$. Bottom: $|E_z(z)|$ at $x=0$ for the cases $\Lambda =0$ (blue solid line), 3 (black dashed line), and 100 (magenta dotted line). See following Eq. (8) for definitions of dimensionless $x$ and $z$.

Figure 3

Self-consistent sheath voltage on a log scale and (inset) linear scale for ${\Lambda }_0<10$. The solid blue line is the numerical result and the magenta dots are the asymptotic results. Note the different scales for both axes in the inset.