In this study, we employ the multi-scale and multi-physics methodology known as field emission coupled with molecular dynamics (FEcMD) simulations to elucidate the thermal runaway of Cu nano-protrusion under the various radiofrequency E-fields applied, unraveling the probable relationships between thermal runaway onset time and other macroscopic electrical field parameters including frequency and field amplitude. The results demonstrate that the conical Cu nanotips could undergo a geometrical transition from the initial conical apex to a more blunted mushroom head-like shape under the intense Joule and Nottingham heating processes with the applied high radiofrequency E-fields, resembling the structural evolutions of the same nanotips under the static high E-field. Most importantly, we predict that the thermal runaway onset time shows a monotonical decreasing with the increase of E-field amplitude, while it exhibits an unusual quartic-like relationship with the variation of the E-field frequency in a range from 1 GHz to 100 GHz. The non-monotonical change of onset time for thermal runaway process versus the frequency is suggested to be closely related to the dedicated competition between the energy dissipation process mediated by the finite energy exchange rate between phonons and electrons and the Joule heating mechanism due to the intense field electron mission under high E-field amplitude.