Submesoscale processes with spatiotemporal scales of 0.1~10 km and hours to 1 day play an important role in the ocean energy cascade and material transport. Up to now, most studies focusing on submesoscales are based on high-resolution numerical simulations, since observations are scarce due to a limitation of their small scales. However, numerical studies on submesoscales deeply depend on the choice of sub-grid turbulence closures that are of significance in determining the simulation performance. Here, submesoscale processes generated at an idealized mesoscale eddy are investigated based on a 1-km resolution Regional Ocean Modelling System (ROMS) model, and the model performance in terms of submesoscales is analyzed and emphasized with three popular turbulence closure schemes, K-Profile Parameterization (KPP), Mellor-Yamada (MY) and General length-scale closure schemes (GLS). The analysis shows that the magnitude of the submesoscale kinetic and potential energy follows: GLS>MY>KPP, despite a general consistent variation by the three closures. To figure out the dominant factors controlling the submesoscale energy budget, three terms including the baroclinic term (BC), the barotropic term (BT), and the background to submesoscale potential energy conversion term (BSP) are analyzed. The results show that the BC is the source of submesoscale kinetic energy, while the BT tends to transfer submesoscale kinetic energy to mesoscales inversely, and the BSP is the submesoscale potential energy source. In comparison, the GLS simulates the largest BC, BT, and BSP. Overall, the order of the BC and BSP is: GLS>KPP>MY, and the order of the BT is: GLS>MY>KPP.