## 4 Results  
### Case 1 and Case 2: Same Subduction Rates, Different Overriding Plate Movement Rates  
Figure 4 depicts the thermal structure of the subduction zone located beneath the Halmahera volcanic arc. This model was created with a constant subduction rate \(V_{0}\) (around 3 cm/a), but with different rates for the movement of the overriding plate \(U\). Specifically, Case 1 involves an overriding plate movement rate \(U_{1}\) of approximately 6 cm/a, whereas Case 2 has a rate \(U_{2}\) of about 4 cm/a. The results indicate that a high-temperature wedge zone developed beneath the Halmahera volcanic arc due to the westward movement of the plate. In Case 1, where the overriding plate movement rate significantly exceeded the subduction rate (\(U_{1}>V_{0}\)), a prominent high-temperature geosynclinal area emerged. Extended periods of rapid convergence could result in unusually high temperatures (Figure 4b1) at depths ranging from 10 to 20 km, potentially reaching the melting point of hydrated mantle rock, which could affect arc magmatism. On the other hand, in Case 2, where the rate of movement of the overriding plate was only slightly higher than the subduction rate (\(U_{2}>V_{0}\)), the subduction zone beneath the Halmahera volcanic arc, characterized by lower temperatures, expanded. Over time, the temperature within the mantle wedge decreased (refer to Figure 5c1), leading to a gradual decline in arc magmatism. Comparison of Case 1 and Case 2 suggests that when the subduction rate remains constant, a faster movement rate of the overriding plate promotes higher temperatures in the mantle wedge. This increase in temperature favors the partial melting of mantle material, leading to enhanced arc magmatic activity. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Parameters** & **Mantle** & **Sediments** & **Ocean Crust** & **Weak Zone** & **Upper Crust** & **Lower Crust** \\ \hline Space (km) & 85 & 5 & 10 & 15 & 20 & 15 \\ Thermal conductivity (W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\)) & 3.3 & 0.85 & 2.5 & 2.5 & 2.5 & 2.5 \\ Specific heat capacity (J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\)) & 750 & 750 & 750 & 750 & 750 \\ Density (kg\(\cdot\)m\({}^{-3}\)) & 3100 & 2300 & 2900 & 2900 & 2700 & 2800 \\ Thermal expansion (K\({}^{-1}\)) & 2 \(\times\) 10\({}^{-5}\) & 2 \(\times\) 10\({}^{-5}\) & 2 \(\times\) 10\({}^{-5}\) & 2 \(\times\) 10\({}^{-5}\) & 2 \(\times\) 10\({}^{-5}\) & 2 \(\times\) 10\({}^{-5}\) \\ Angles of internal friction (\({}^{\circ}\)) & 20 & 5 & 10 & 2 & 20 & 20 \\ Cohesions (Pa) & 20 \(\times\) 10\({}^{6}\) & 1 \(\times\) 10\({}^{6}\) & 1 \(\times\) 10\({}^{6}\) & 1 \(\times\) 10\({}^{6}\) & 20 \(\times\) 10\({}^{6}\) & 20 \(\times\) 10\({}^{6}\) \\ Grain size exponents & 3 & 1 & 1 & 1 & 1 & 1 \\ \hline \end{tabular}
\end{table} 
Table 2: Physical and Geometrical Parameters of the model. Figure 4: The effect of the overriding plate movement rate \(U\) on the evolution of thermal structure. (**a**) The current thermal structure, with units in \({}^{\circ}\)C; (**b**) Case 1: the thermal structure after 1 Myr at a higher overriding plate movement rate \(U_{1}\), also in \({}^{\circ}\)C, where (**b1**) shows the evolution of the thermal structure above the subducted slab over 0 to 5 Myr at the higher rate \(U_{1}\); (**c**) Case 2: the thermal structure after 1 Myr at a lower overriding plate movement rate \(U_{2}\), with units in \({}^{\circ}\)C, where (**c1**) depicts the thermal structure evolution above the subducted slab, over 0 to 5 Myr, at the lower rate \(U_{2}\). Figure 5: The effect of the subduction rate \(V\) on the evolution of thermal structure.