  The linear and nonlinear characteristics of the electron cyclotron drift
instability (ECDI) have been studied through the particle-in-cell (PIC) and
continuum Vlasov simulation methods in connection with the effects of the
azimuthal length (in the $E \times B$ direction) on the simulations. Simulation
results for a long azimuthal length (17.82 cm $= 627\;v_d/\omega_{ce}$, where
$\omega_{ce}$ is the electron cyclotron frequency and $v_d$ is the $E\times B$
drift of the electrons) are reported, for which a high resolution is achieved
in Fourier space. For simulations with a long azimuthal length, the linear
growth rates of the PIC simulations show a considerable discrepancy with the
theory, whereas the linear growth rate of the Vlasov simulations remains close
to the theory. In the nonlinear regime, the inverse cascade is shown in both
PIC and Vlasov simulations with a sufficiently large azimuthal length. In
simulations with a short azimuthal length, however, the inverse cascade is
barely observed. Instead, the PIC simulations with a short azimuthal length
(0.5625 cm $=19.8\;v_d/\omega_{ce}$) show an essentially continuous nonlinear
dispersion, similar to what is predicted by the ion-sound turbulence theory. It
is shown that, in the PIC and Vlasov simulations, the inverse cascade coincides
with the formation and merging of electron structures in phase space. This
process, however, terminates differently in the PIC simulations compared with
the Vlasov simulations. Larger amplitudes of ECDI fluctuations are observed in
the PIC simulations compared with the Vlasov simulations, leading to an
intensified electron heating and anomalous current. This suggests that the
statistical noise of PIC simulations might contribute to the extreme electron
heating that has been observed in previous studies.
