Control of Coherent Structures via External Drive of the Breathing Mode

Abstract

The Hall thruster exhibits two types of large-scale coherent structures: axially propagating breathing mode (m = 0) and azimuthal, with low m, typically m = 1, spoke mode. In our previous work, it was demonstrated that axial breathing mode can be controlled via the external modulations of the anode potential. Two regimes of the thruster response, linear and nonlinear, have been revealed depending on the modulation amplitude. In this work, using the high-speed camera images and developed image-processing technique, we have investigated the response of the azimuthal mode to the external modulations. We have found that, in linear regime, at low modulation voltages, axial and azimuthal structures coexist. At larger amplitudes, in the nonlinear regime, the azimuthal mode is suppressed, and only axial driven mode remains.

Appendices

APPENDIX

DETAILS OF THE IMAGE PROCESSING TECHNIQUES

Recorded video can be considered as a 3D matrix of image pixel brightness as a function of position and time,

$$I = I(x,y,t),$$

where x and y are pixel column and row, respectively, and t is a time index. There are three main components of the video, which can be identified by looking at it. The first component is the image from the thruster channel itself and the background plasma, which gives some DC level of intensity. The second is oscillations of the intensity of the whole image due to the breathing mode, or m = 0 mode. The third is the azimuthal oscillations of the intensity due to the rotating spoke, or m = 1 mode, which are of the main interest. To filter these oscillations from the DC level and m = 0 mode several steps were done. Stages of image processing are shown in Fig. 15.

Fig. 15.

Image processing stages at one time step. From the left to the right: raw image, cut by R_ext, removed background, applied mask and removed effect from the breathing mode, and image in polar coordinates.

The scheme of image processing includes the following steps.

1. Find external radius R_ext of the channel and the anode cup radius R_int. This is done by the MATLAB function “imfindcircles.”

2. Cut frames by the channel size.

3. Calculate mean DC level of all frames over time to obtain the mean image M, defined as

$$M(x,y) = \frac{1}{{{{N}_{t}}}}\mathop \sum \limits_t I(x,y,t).$$

4. Subtract M(x, y) matrix from each frame. Therefore, by subtracting it from each frame the background intensity is eliminated,

$${{I}_{{{\text{AC}}}}}\left( {x,y,t} \right) = I\left( {x,y,t} \right) - M(x,y).$$

5. Apply circular mask on each frame by setting pixel values to zero within R_int and outside R_ext.

6. Split each frame on circles C_it with radius r_i and width of one pixel, so R_int ≤ r_i ≤ R_ext, and obtain R_ext − R_int arrays of pixels.

7. Calculate average intensity of each circle as

$${{c}_{i}} = \frac{1}{{{{N}_{c}}}}\mathop \sum \limits_\theta {{C}_{{it}}}(\theta ).$$

8. Eliminate breathing oscillation effect from each circle by subtracting average intensity c_i from each C_it array.

9. Transform each frame to polar coordinates,

$$I\left( {x,y,t} \right) \to I(R,\theta ,~t).$$

10. Because our analysis is focused on identifying the rotational structures, the radial dependence will be neglected. This is accomplished by average pixel intensity along the radius. Therefore, the matrixes I_avg(θ, t) are obtained,

$${{I}_{{{\text{avg}}}}}\left( {\theta ,~t} \right) = \frac{1}{{{{R}_{{{\text{ext}}}}} - {{R}_{{{\text{int}}}}}}}\mathop \sum \limits_{r = {{R}_{{{\text{int}}}}}}^{{{R}_{{{\text{ext}}}}}} I(r,\theta ,~t).$$

11. Such matrices can be combined along time dimension in so-called 2D “spoke surface.” This surface represents variation of pixel intensity in angular direction θ and with time t.

12. In order to determine characteristic frequencies of azimuthal modes, the 2D Fourier transform was applied to it. The particular implementation is based on the built-in “fft” command in MATLAB. Results are shown in Fig. 16.

Fig. 16.

(a) Spoke surface. The angled lines correspond to the spoke propagating in azimuthal direction with time. (b) 2D FFT matrix of the spoke surface. This matrix contains a discrete signal for each mode number. One can see that the m = 0 mode is still present even though it was cleaned out of the video. This happens because the breathing mode is much stronger than the azimuthal modes (m = 1).

13. Breathing mode was identified through the Fourier transform of the signal, which was defined as follows:

$$B\left( t \right) = \mathop \sum \limits_{x,y} I\left( {x,y,t} \right).$$