Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem

The purpose of this article is three-fold. The first is to investigate the linear stability of the basic flow governed by the three-dimensional (3D) quasi-geostrophic (QG) equation with a steady wind force. We show that the basic flow will become unstable as the Reynolds number crosses a critical value. The second is to study the dynamic transition associated with the instability of the basic flow at the critical Reynolds number. It is proved that the type of the phase transition is determined by a parameter called transition number which is mainly determined by the Reynolds number and the aspect ratio. More precisely, the negative (positive) sign of the real part of the transition number corresponds to the continuous (catastrophic) transition. The last objective is to apply numerical experiments to examine the specific transition type for specified control parameters. We show that the real part of the transition number is always negative, leading to only the continuous transition. (C) 2019 Elsevier Inc. All rights reserved.