## 3 Results and Discussions
### Wall Erosion When the Value of \(\alpha\) Is Positive  
When the value of \(\alpha\) is positive, the magnetic field line is convex to the anode, which will lead to a change in the electric potential distribution compared to the case with the electric field \(\mathbf{E}\) perpendicular to the magnetic field \(\mathbf{B}\). Since \(\mathbf{B}_{y}\) and \(\mathbf{E}_{z}\) are preset and remain unchanged in all cases, the periodic instability structure caused by \(\mathbf{B}_{y}\)\(\times\)\(\mathbf{E}_{z}\) in the simulated region can always be observed and only the wavelength in the simulation area corresponding to different \(\alpha\) varies, as shown in Figure 7. When \(\alpha=15^{\circ}\), the value of \(\mathbf{B}_{z}\) is small and exerts little impact on the ECDI. Thus, the wavelength in the simulated area is the same as the situation where the electric field \(\mathbf{E}\) is perpendicular to the magnetic field \(\mathbf{B}\). However, with the increase in \(\alpha\), the wavelength of ECDI decreases. In addition, as can be seen by comparing Figures 4 and 7, the potential on the wall with positive \(\alpha\) is lower than that of the case with \(\mathbf{E}\bot\mathbf{B}\), and the wall potential decreases as the value of \(\alpha\) increases. Figure 6: The erosion rate of the dielectric wall along the azimuthal direction in different time periods with \(\mathbf{E}\bot\mathbf{B}\). The behavior of electrons could be the explanation for the above phenomenon. The periodic distribution of the potential driven by ECDI leads to an electric field \(\widetilde{\mathbf{E}}_{x}\) along the azimuthal direction, as shown in Figure 5. The average value of \(\widetilde{\mathbf{E}}_{x}\) is non-zero and positive. When the magnetic field component \(\mathbf{B}_{z}\) is applied in the axial direction, a new orthogonal field \(\widetilde{\mathbf{E}}_{x}\times B_{z}\) will be generated. The new orthogonal field induces a new drifting motion perpendicular to the wall surface. If the value of \(\alpha\) is positive, the new drifting motion of the electrons is towards the wall. This movement increases the velocity of the electrons tending to the wall. Therefore, the electron flux that can reach the wall becomes larger, and the potential drop in the sheath increases. As the value of the positive \(\alpha\) increases, the radial drift velocity increases. More electrons can reach the wall and make the wall potential decreases. Therefore, it can be seen from the electron density distribution shown in Figure 8 that as the value of positive \(\alpha\) increases, the wall potential decreases and the low-density region of the electrons near the wall expands. Figure 7: The 2D spatial distributions of electric potential with different values of the positive \(\alpha\). Although the axial component of the curved magnetic field does not directly affect the ion distribution, the ion behavior will be affected by the electric potential dominated by the electron motion. When the electric field \(E\) is perpendicular to the magnetic field \(B\), most of the ions are captured by the \"ion trapping\" phenomenon induced by ECDI and move primarily in the azimuthal direction within the channel. When the magnetic field line is convex to the exit plane of HETs, the potential of the wall decreases. The potential gradient along the radial direction allows a greater acceleration of the ions moving toward the wall. Figure 9 shows the ion velocity distribution function for the radial component with different values of the positive \(\alpha\). In order to eliminate the interference of the quasi-neutral region on the particle velocity, all ions within \(10\lambda_{D\varepsilon}\) from the upper boundary are neglected in the statistics. It can be seen that as the value of the positive \(\alpha\) increases, the average velocity of ions toward the wall increases, and the distribution of the ion velocity becomes more uniform. Figure 8: The 2D spatial distribution of electron density with different values of the positive \(\alpha\).