  Electromagnetic radiation at higher harmonics of the plasma frequency
($\omega \sim n\omega_{pe}, n > 2$) has been occasionally observed in type II
and type III solar radio bursts, yet the underlying mechanism remains
undetermined. Here we present two-dimensional fully kinetic electromagnetic
particle-in-cell simulations with high spectral resolution to investigate the
beam-driven plasma emission process in weakly magnetized plasmas of typical
coronal conditions. We focused on the generation mechanisms of high-harmonic
emission. We found that a larger beam velocity ($u_d$) favors the generation of
the higher-harmonic emission. The emissions grow later for higher harmonics and
decrease in intensity by $\sim$2 orders of magnitude for each jump of the
harmonic number. The second and third harmonic ($\rm H_2$ and $\rm H_3$)
emissions get closer in intensity with larger $u_d$. We also show that (1) the
$\rm H_3$ emission is mainly generated via the coalescence of the $\rm H_2$
emission with the Langmuir waves, i.e., $\rm H_2 + L \rightarrow H_3$, wherein
the coalescence with the forward-propagating beam-Langmuir wave leads to the
forward-propagating $\rm H_3$, and coalescence with the backward-propagating
Langmuir wave leads to the backward-propagating $\rm H_3$; and (2) the $\rm
H_4$ emission mainly arises from the coalescence of the $\rm H_3$ emission with
the forward- (backward-) propagating Langmuir wave, in terms of $\rm H_3 + L
\rightarrow H_4$.
