In this paper， a LOG capable of operating from the 4^th to the 9^th harmonic is designed and investigated. With the assistance of the 3-D particle-in-cell simulation， the key features of the designed LOG， beam-wave interaction dynamics and high-harmonic operation regimes are studied. It is shown that by tuning the external magnetic field intensity， successive excitation of the oscillation at a number of neighboring harmonics can be achieved， corresponding to radiation frequencies between 240 GHz to 460 GHz with the maximum radiation power of 19 kW. Then， a detailed study on the competitions among the 7^th， 8^th and 9^th harmonics are conducted， providing an insight into the multi-modes， multi-harmonics behavior of the high-harmonic LOG. Following that， the methods to stabilize the beam-wave interaction and enable single mode operation at high harmonics are discussed. Additionally， the characterization of the ohmic loss power at different operation harmonics is conducted.