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 quasiperiodic coherent structures from the single tower observation. Analysis of turbulence fluxes and anisotropy demonstrates the mechanism, that is, 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. Plain Language Summary The turbulence intensity is always influenced by submesoscale motions in a nocturnal stable boundary layer. This multiscale interaction can be displayed by the hockey-stick transition, which is depicted by the "local and global shear" assumption. Although this pattern is noticed widely in many different field observations, the intrinsic mechanism of averaged flow influences turbulence kinetic energy via shear and buoyancy is unclear. Because during the interaction, turbulent eddies in the nocturnal stable boundary layer are embedded in the coherent structures, leading to the difficulty in the estimation of the time length of the interaction. In this research, we deploy the Koopman operator to expose invariant subspaces of the Ri series to identify the quasiperiodic coherent structures from the single tower observation. Anisotropy analysis based on the time length-averaged result from the subspaces reveals the existence of the mechanism, that is, horizontal kinetic energy transport coupled with a vertical downward flux, which causes the unique evolution of turbulence anisotropy. Moreover, a dynamical transition is discovered when the anisotropy invariants change with kinetic energy which determines the threshold of HOST by a group of quadratic curves about the shear and vertical momentum when the anisotropy crossed over the transition point.

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 quasiperiodic coherent structures from the single tower observation. Analysis of turbulence fluxes and anisotropy demonstrates the mechanism, that is, 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. Plain Language SummaryThe turbulence intensity is always influenced by submesoscale motions in a nocturnal stable boundary layer. This multiscale interaction can be displayed by the hockey-stick transition, which is depicted by the "local and global shear" assumption. Although this pattern is noticed widely in many different field observations, the intrinsic mechanism of averaged flow influences turbulence kinetic energy via shear and buoyancy is unclear. Because during the interaction, turbulent eddies in the nocturnal stable boundary layer are embedded in the coherent structures, leading to the difficulty in the estimation of the time length of the interaction. In this research, we deploy the Koopman operator to expose invariant subspaces of the Ri series to identify the quasiperiodic coherent structures from the single tower observation. Anisotropy analysis based on the time length-averaged result from the subspaces reveals the existence of the mechanism, that is, horizontal kinetic energy transport coupled with a vertical downward flux, which causes the unique evolution of turbulence anisotropy. Moreover, a dynamical transition is discovered when the anisotropy invariants change with kinetic energy which determines the threshold of HOST by a group of quadratic curves about the shear and vertical momentum when the anisotropy crossed over the transition point.

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