The hockey-stick transition (HOST), which is depicted by the ‘local and global shear’ assumption, about the turbulence kinetic energy with the averaged flow intensity is noticed widely. However, the intrinsic mechanism of averaged flow influences turbulence kinetic energy via shear and buoyancy is missing. In this research, we deploy the Koopman operator to expose invariant subspaces of the Ri series to identify the quasi-periodic coherent structures from the single tower observation. Analysis of turbulence fluxes and anisotropy demonstrates the mechanism, i.e. horizontal kinetic energy coupled with vertical downward flux, whereby anisotropy evolution is changed. Examining the anisotropy invariants changing with kinetic energy reveals a dynamical transition that determines the threshold of HOST. Moreover, the mechanism about how the shear and vertical momentum influences the transition point of HOST is at first given by a group of a quadratic relationship when the anisotropy crossed over the transition point