  In spherical tokamaks, the electron plasma frequency is greater than the
electron cyclotron frequency. Electromagnetic waves in the electron cyclotron
range of frequencies are unsuitable for directly heating such plasmas due to
their reduced accessibility. However, mode-conversion of the extraordinary wave
to the electron Bernstein wave (X-B mode-conversion) at the upper hybrid
resonance makes it possible to efficiently couple externally-launched
electromagnetic wave energy into an overdense plasma core. Traditional
mode-conversion models describe an X-mode wave propagating in a potential
containing two cutoffs that bracket a single wave resonance. Often, however,
the mode-conversion region is in the edge, where turbulent fluctuations and
blobs can generate abrupt cutoffs and scattering of the incident X-mode wave.
We present a new framework for studying the X-B mode-conversion which makes the
inclusion of these fluctuations analytically tractable. In the new approach,
the high-field cutoff is modelled as an infinite barrier, which manifests as a
boundary condition applied to a wave equation involving only one cutoff
adjacent to the resonance on the low-field side. The new model reproduces the
main features of the previous approach, yet is more suitable for analyzing
experimental observations and extrapolating to higher dimensions. We then
develop an analytical estimate for the effect of small-amplitude,
quasi-monochromatic density fluctuations on the X-B mode-conversion efficiency
using perturbation theory. We find that Bragg backscattering of the launched
X-mode wave reduces the mode-conversion efficiency significantly when the
fluctuation wavenumber is resonant with the wavenumber of the incident X-mode
wave. These analytical results are corroborated by numerically integrating the
mode-conversion equations.
