Stabilization mechanism of the feedback instability with height-resolved ionosphere and Alfvén resonator models

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^ > 3:3 with ^ in ¼ 3. In the case of ^ in ¼ 5,

ReðzÞ ’ 1 for ReðxÞ

where the high-frequency modes are completely stabilized due to the

strong inhomogeneity, the frequency range providing ReðzÞ ’ 1

^ > 2:5. This is because

extends to the lower frequency regime of ReðxÞ

the real part of the denominator in the integral in Eq. (20) vanishes at

a lower frequency for the stronger inhomogeneity with the smaller

minimum of l

^ P ðzÞ [see Eqs. (7) and (14)].

^ is obtained in Eq. (21), one

Once the functional form of zðxÞ

^ including the

can solve the dispersion relation in Eq. (20) with zðxÞ,

effects of ionospheric inhomogeneity. Namely, the set of Eqs. (12) and

(13) with  in ¼ 0 and Rrec ¼ 0 is solved for the magnetosphere but

with the height-averaged ionosphere model using Eqs. (20) and (21).

Figure 6 displays the real and imaginary parts of the eigenfrequency

given in Eq. (12) with lP ¼ Rrec ¼ 0 and F ¼ 1 coupled with Eqs.

(13), (20), and (21), where the results agree fairly well with those in

Figs. 2(c)–2(h). Thus, it is concluded that the stabilization of highfrequency modes in the case with the ionospheric inhomogeneity is

attributed to changes of the effective impedance ð1  zÞ.

The present results showing stabilization of the feedback instability for the high-frequency modes related to IAR are qualitatively consistent with the simulation results in the earlier work.13 Here, it is also

noteworthy that stabilization of the low-frequency modes was not

shown by the numerical simulation.13 In contrast, the present study

verifies that the low-frequency modes remain unstable in the M–I coupling model, which is explained in terms of the frequency dependence

of the effective impedance (1  z).

ARTICLE

Physics of Plasmas

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T.-H. Watanabe and S. Maeyama, “Unstable eigenmodes of the feedback instability with collision-induced velocity shear,” Geophys. Res. Lett. 45,

10043–10049, https://doi.org/10.1002/2017GL073415 (2018).

ARTICLE

scitation.org/journal/php

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A. Potapov, T. Polyushkina, B. Dovbnya, B. Tsegmed, and R. Rakhmatulin,

“Emissions of ionospheric Alfven resonator and ionospheric conditions,”

J. Atmos. Sol.-Terrestrial Phys. 119, 91–101 (2014).

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T.-H. Watanabe (2022). “A dispersion solver for the feedback instability with

inhomogeneous ionosphere and ionospheric Alfven resonator,” Github. https://

doi.org/10.5281/zenodo.7023952

18 October 2023 06:44:06

Phys. Plasmas 30, 042903 (2023); doi: 10.1063/5.0139084

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