Zonal flows and, more generally, zonal structures are known to play important self-regulatory roles in the dynamics of microscopic drift-wave-type turbulences. Since toroidal Alfvén eigenmode (TAE) plays crucial roles in the Alfvén wave instabilities in burning fusion plasmas, it is, thus, important to understand and assess the possible roles of zonal flow and structures on the nonlinear dynamics of TAE. It is shown that zonal flow or structure spontaneous excitation is more easily induced by finite amplitude TAEs including the proper trapped-ion responses, causing the zonal structure to be dominated by the zonal current instead of the usual zonal flow. This work shows that proper accounting for plasma equilibrium geometry as well as including kinetic thermal ion treatment in the nonlinear simulations of Alfvénic modes are important ingredients for realistic comparisons with experimental measurements, where the existence of zonal fields has been clearly observed.