0.7, respectively. In the case of sin $^{2}$2 θ$_{13}$ = 10 − $^{4}$, P ( E$_{ν}$ ) and ¯ P ( E$_{ν}$ ) are about 0.2 and 0.4 in t = 0 − 2 s, respectively. These survival probabilities indicate intermediate adiabaticity between the cases for large θ$_{13}$ and small θ$_{13}$ . At the later times they become large as shown in the large θ$_{13}$ case. 

We can understand these behaviors considering the shock wave propagation. The adi- abaticity of H-resonance is estimated by γ in Eq. (7). When the shock wave reaches H-resonance region, the adiabaticity changes. If sin $^{2}$2 θ$_{13}$ is large and there is not the shock wave at the resonance region, γ is larger than 1, and the resonance is adiabatic. As a result, the survival probabilities are 0. When the shock wave reaches a resonance, large density gradient reduces γ and the resonance becomes non-adiabatic. Thus, the survival probabili- ties become finite during the shock passage in the resonance region. In contrast, if sin $^{2}$2 θ$_{13}$ is very small, γ is smaller than 1, and the resonance is non-adiabatic. Consequently, the survival probabilities are finite, and the influence of the shock wave hardly appears to the survival probability, since H-resonance totally in non-adiabatic regardless the existence of the shock wave. In the case of sin $^{2}$2 θ$_{13}$ = 10 − $^{2}$, however, the influence of the shock wave is not clearly seen though the sin $^{2}$2 θ$_{13}$ is large. This is because that sin $^{2}$2 θ$_{13}$ is very large and γ is larger than 1 even when | 1 n$_{e}$ dn$_{e}$ dr | becomes large. 

γ can be rewritten by the ratio of the length characterizing the neutrino oscillations L$_{osc}$ = 2 E$_{ν}$ Δ ˜ m $^{2}$sin 2 ˜ $_{θ}$, where Δ ˜ m 2 = { $_{(2}$√$_{2}$$_{G}$$_{F}$$_{E}$$_{ν}$$_{n}$$_{e}$$_{(}$$_{r}$$_{)}$$_{/}$$_{Δ}$$_{m}$ 2 − cos 2 θ ) 2 + sin $^{2}$2 θ } 1 / $^{2}$Δ m $^{2}$, to the length of the level crossing region, i.e. the level crossing length δr = sin 2 θ cos 2 θ | 1 n$_{e}$ dn$_{e}$ dr | − 1 at resonance. Therefore, the adiabaticity of the neutrinos can be evaluated by L$_{osc}$ and δr [59]. The resonance is adiabatic when the level crossing length is larger than the oscillation length, L$_{osc}$ ≪ δr . Figure 3 shows the level crossing region and the oscillation length of neutrinos ( E$_{ν}$ = 5 MeV) in the case of inverted mass hierarchy as a function of the radius. The left panels of Figure 3 are at t = 0 s, and the right panels are at t = 3 s, respectively. Shown are the results for sin $^{2}$2 θ$_{13}$ = 10 − $^{2}$, 10 − $^{3}$, 10 − 4 and 10 − 5 from top to bottom. Red and green lines display the oscillation lengths of ν$_{e}$ and ¯ ν$_{e}$ , and blue line shows the level crossing length using θ$_{13}$ . The oscillation length of ¯ ν$_{e}$ becomes extremely large on the resonance, and its behavior depends on sin $^{2}$2 θ$_{13}$ below or above the resonance strongly. On the other hand, the oscillation length of ν$_{e}$ changes rather gently as a function of radius, and strong dependence on sin $^{2}$2 θ$_{13}$ is not clearly seen. As sin $^{2}$2 θ$_{13}$ becomes smaller or the shock wave reaches the resonance, the level crossing length becomes shorter. 

