“…While the observations of monochromatic/broadband toroidal waves have been reported by both event studies (Matsui et al, 2007;Sarris, 2014;Sarris et al, 2009;Shi et al, 2020;Vellante et al, 2004) and statistical studies (Liu et al, 2009;Murphy et al, 2020;Takahashi et al, 2014;Takahashi, Hartinger et al, 2015), toroidal waves with multi-harmonic discrete frequencies have been mainly investigated in event studies (Engebretson et al, 1986(Engebretson et al, , 1987Takahashi, Denton et al, 2015;Takahashi, Hughes et al, 1984;Takahashi, Waters et al, 2015;Takahashi et al, 2020). The multi-harmonic toroidal waves (MTWs hereinafter) are considered to be standing Alfvén waves with multiple eigenmodes because the wave frequency of MTWs decreases (increases) as the L-value increases (decreases) (e.g., Engebretson et al, 1986).…”