We examined the influence of internal noise on the irregular switching of the shape of the free surface of fluids in an open cylindrical vessel driven by a bottom disk rotating at constant speed [Suzuki, Iima, and Hayase, Phys. Fluids 18, 101701 (2006)PHFLE61070-663110.1063/1.2359740]. A slight increase in the disk-rim gap (less than 3% of the disk radius) was established experimentally to cause significant changes in this system, specifically, frequent appearance of the surface descending event connecting a nonaxisymmetric shape in strong mixing flow (turbulent flow) and an axisymmetric shape in laminar flow, as well as a shift in critical Reynolds number that define the characteristic states. The physical mechanism underlying the change is analyzed in terms of flow characteristics in the disk-rim gap, which acts as a noise source, and a mathematical model established from measurements of the surface height fluctuations with noise term.