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Alfvén waves play an important role in coronal heating and solar wind acceleration. Despite the recent progress of theoretical and observational studies on Alfvén waves, their dissipation processes have not been fully understood. Parametric Decay Instability (PDI) is a coupling process between waves in which a large-amplitude, forward-propagating Alfvén wave resonates and decays into a backward-propagating Alfvén wave and a forward-propagating slow magnetosonic wave. The slow magnetosonic waves generated by PDI contribute to the efficient heating of the solar wind ions through the formation of shock waves. Alfvén turbulence enhanced by Alfvén waves reflected by the density fluctuations contributes to the turbulent heating.[e.g., Shoda et al., 2018]. Therefore, PDI is an important physical process in coronal heating and solar wind acceleration. Recent studies have advanced our understanding of PDI in the solar atmosphere. Theoretical and observational studies showed that PDI operates in the solar wind [e.g., Tenerani and Velli, 2013; Bowen et al., 2018], and in the near-transition layer of the lower solar atmosphere [e.g., Hahn et al., 2022]. Previous studies have shown that the growth rate of PDI increases with higher temperature perpendicular to the magnetic field than parallel, lower plasma beta parallel to the magnetic field, and larger amplitude of the parent wave [e.g., Tenerani et al., 2017].
In this study, using the dispersion relation derived from the linearized CGL equations, we evaluated the linear growth rate of PDI in the solar corona. We investigated the effect of the expansion of plasma volume to PDI quantitatively. In the computation of the linear growth rate, we used the parameter representing the temperature ratio between parallel and perpendicular to the background magnetic field (${\xi}$), the plasma beta computed by the parallel component of the plasma pressure (${\beta_\parallel}$), and the amplitude of the parent wave normalized by the background magnetic field intensity ($\hat{B_\perp}$ = ${B_\perp}$ / ${B_0}$). To incorporate the effect of expansion, we expressed physical quantities in a form proportional to a power of the radial distance R from the Sun. By using these parameters, we calculated ${\xi}$, ${\beta_\parallel}$, and $\hat{B_\perp}$ at positions of 10,000 km, R = $\sqrt{2}$×10,000 km, and R = 20,000 km. We set the position of the lower boundary of the solar corona as R = 10,000 km and (${\xi}$, ${\beta_\parallel}$, $\hat{B_\perp}^2$) as (3, 0.01, 0.01) in this region, which is determined by Huang et al.,2023 and Gary et al., 2001. Then we calculated the linear growth rate.
Results have shown that the maximum growth rate is 0.1131${\omega_{0}}$ at R = 10000km, 0.124${\omega_{0}}$ at R = $\sqrt{2}$×10,000 km, and 0.128${\omega_{0}}$ at R =20000km, where ${\omega_{0}}$ is the frequency of the parent wave.
It is the opposite trend previous studies reported that expansion suppresses PDI [e.g., Tenerani et al.,2013, Shoda et al.,2018]. In previous studies, to consider expansion, a term representing the effect of expansion was added to the basic equation, incorporating both the stretching in the r direction (acceleration) and the stretching perpendicular to the r direction (expansion). On the other hand, in the present study, the expansion perpendicular to to the r direction without the acceleration is considered. Our results suggest that the acceleration may play an important effect in suppressing PDI.
