4.3 Numerical Analysis
4.3.3 Effect of variation of angle of propagation with finite electron beam
Furthermore, the coupling between mode 2 and mode 6 still remains intact and happens at 0.00007. However, the coupling between modes 2 and 5 is weak as compared to that in case ofθ =15◦ (see Fig. 4.2c) and happens in the kλDe range 0.00009-0.00015. The slow ion acoustic mode (mode 4) completely decouples from the rest of the modes.
In Fig. 4.3b, the dispersion relation curve is plotted forθ =40◦andUcb
s =0. It is seen that the mode 3 (fast ion acoustic) and mode 4 (slow ion acoustic) do not couple with any of the cyclotron modes. The only coupling is between mode 2 and mode 6 atkλDe 0.000070 and then between mode 2 and mode 5 in thekλDerange 0.0001-0.000180. Further increasing the angle of propagation weakens the coupling between mode 2 and mode 5 (this has been tested forθ =60◦, but not shown here).
With further increase in angle of propagation to 80◦, the coupling between mode 2 and mode 5 also vanishes with only coupling existing between mode 2 and mode 6. A notable thing to mention here is that the frequencies of slow and fast ion acoustic mode decreases substantially compared with that of Fig. 4.3b.
Finally forθ =90◦, the only cyclotron mode exist in the system as seen in Fig. 4.3d.
4.3.3 Effect of variation of angle of propagation with finite electron
Figure 4.4: Plot of real frequency,ωr/ωpeand growth rate,ωi/ωpe versuskλDeforθ =0◦ andUcb
s =4.5. Other parameters are the same as in Fig. 4.1
in two different panels because the merging and coupling of modes which is happening oc- curs at differentkλDe scale and it is not be possible to show both coupling and merging in same graph. In Fig. 4.5a, it can be seen that the coupling happening at 6 different places in the positive y-axis region and 4 different places in negative y-axis region. This is in sharp contrast to what was reported for in Sect. 4.3.2, where coupling was same in both the axis.
Fig. 4.5b shows the same dispersion plot as in Fig. 4.5a, but in the higherkλDerange. Here, one can clearly see the merging of mode 11 with mode 6 thereby generating a unstable re- gion first in thekλDe range 0.0264-0.0267 with maximum growth rate 1.740e-08 and then in 0.0485-0.0493 with maximum growth rate 7.670e-08. The first merging happens between mode 11 and mode 6 and then with mode 5. Thus, the proton and helium cyclotron modes are excited by an electron beam.
With further increase in angle of propagation to 10◦ (not shown here), mode 11 still merges with mode 6 inkλDerange 0.0251-0.0255 with maximum growth rate 3.514e-08 and then with mode 5 in thekλDerange 0.0429-0.0443 with maximum growth rate 16.062e-08.
The dispersion relation curves forθ =30◦is shown in Fig. 4.6. As one can see from Fig. 4.6a, the mode 4 does not couple with any other modes. Also, in the negative y-axis, all the coupling regions comes closer to each other. Here, the mode 11 first merges with
(b)
Figure 4.5: Plots showing coupling (Fig. 4.5a) and merging (Fig. 4.5b) of different modes forθ =5◦and Ucb
s =4.5. Other parameters are the same as in Fig. 4.1
(a)
(b)
Figure 4.6: Plots showing coupling (Fig. 4.6a) and merging (Fig. 4.6b) of different modes forθ =30◦and Ucb
s =4.5. Other parameters are the same as in Fig. 4.1
(a)
(b)
Figure 4.7: Plots showing coupling (Fig. 4.7a) and merging (Fig. 4.7b) of different modes forθ =40◦and Ucb
s =4.5. Other parameters are the same as in Fig. 4.1
mode 4 in thekλDe range 0.0195-0.0204 with maximum growth rate 1.0045e-07 and then with mode 6 in the kλDe range 0.0286-0.0310 with maximum growth rate of 4.912e-07 as shown in Fig. 4.6b. The growth rate of merged mode of 11 and 6 increases as compared to that ofθ =5◦ (as shown in Fig. 4.5b) and those of 10◦ (not shown here) and 40◦ (see Fig. 4.7b). Also, forθ =40◦, in negative y-axis, the only coupling happens is between mode 7 and mode 10, which occurs at two differentkλDe range- the coupling between mode 8 &
10, mode 7 & mode 9 and mode 8 & 9 vanishes.
When the angle of propagation is increased to 50◦, the only coupling that exists is between mode 5 & 2 and mode 6 & 2 in the positive y-axis. In the negative y-axis mode 7 couples with mode 10 as shown in Fig. 4.8a. The modes that now merges with mode 11 are mode 4 inkλDe range 0.0164-0.0174 with maximum growth rate to be 12.447e-08 and mode 3 in thekλDerange 0.0227-0.0257 with maximum growth rate to be 61.572e-08. With further increase in the angle of propagation, mode 11 merges with mode 4 and then with 3 with decrease in growth rate and unstable regions shifting to lowerkλDe range. These are electron beam driven slow and fast ion acoustic modes, respectively.
Angle of propagation is increased toθ =80◦and results are shown in Fig. 4.9. Here,
(b)
Figure 4.8: Plots showing coupling (Fig. 4.8a) and merging (Fig. 4.8b) of different modes forθ =50◦and Ucb
s =4.5. Other parameters are the same as in Fig. 4.1
(a)
(b)
Figure 4.9: Plots showing coupling (Fig. 4.9a) and merging (Fig. 4.9b) of different modes forθ =80◦and Ucb
s =4.5. Other parameters are the same as in Fig. 4.1
(a)
(b)
Figure 4.10: Plots showing coupling (Fig. 4.10a) and merging (Fig. 4.10b) of different modes forθ =85◦and Ucb
s =4.5. Other parameters are the same as in Fig. 4.1
it can be seen that there is no coupling between mode 7 and mode 10 (Fig. 4.9a). The growth rate of first merging (mode 11 & 4) and second merging (mode 11 & 3) decreases to 4.588e- 08 and 22.6e-08 respectively (Fig. 4.9b).
When the angle of propagation is further increased toθ =85◦, the coupling between mode 5 and mode 2 also vanishes leaving out the coupling between mode 6 & mode 2 as shown in Fig. 4.10. The merging between modes 11 and 3 and 11 and 4 continue to occur with lower growth rate as compared toθ =80◦.
When the angle of propagation is further increased toθ =87◦, all the modes decouple with mode 11 merging with mode 4 first and then with mode 3 (see Fig. 4.11).
For perpendicular propagation, the acoustic mode vanishes and there exist only proton and Helium cyclotron mode only. At perpendicular propagation there is will not be any unstable region. This is shown in Fig. 4.12.
(b)
Figure 4.11: Plots showing coupling (Fig. 4.11a) and merging (Fig. 4.11b) of different modes forθ =87◦and Ucb
s =4.5. Other parameters are the same as in Fig. 4.1
Figure 4.12: Plot of real frequency,ωr/ωpe versuskλDe forθ =90◦ with Ucb
s =4.5. Other parameters are the same as in Fig. 4.1