Circulation and associated dynamics in the ocean can be favorably depicted by vorticity dynamics in various vorticity equations. Yet, the form and physical interpretation of these equations remain ambiguous. In this study, we analyze the widely-used depth-integrated (DIV; take curl of depth integrated velocity), the depth-averaged (DAV; take curl of depth averaged velocity) and depth-dependent vorticity (DDV; take curl of depth dependent velocity) equations to understand respective physics represented by them. Despite unanimous representation for dynamics of the rotational motion with different mathematic expressions, the physical meaning embedded in these vorticity equations vary and represent different aspects of the circulation dynamics. Besides, as compared with DIV and DAV, DDV is physically more sensible, and DIV and DAV represent the DDV minus/plus Joint Effect of Topography and Velocity (JETV). However, the important flow-topography dynamics can only be reflected by DIV and DAV equations through respective topographic pressure torque (TPT) and Joint Effect Baroclinicity and Relief (JEBAR). We conduct a numerical study to elucidate and consolidate understanding of extrinsic-intrinsic forcing in the semi-closed South China Sea basin. We found that DIV equation, through the domain-integration, can explicitly isolate the extrinsic/intrinsic dynamic circulation. To explicitly display the topography and extrinsic linkage as well as to retain the physically sensible of the equation, an adjusted DIV equation/ Adjusted Layer Integrated Vorticity Equation (ALIVE) should be utilized.