The atmospheric convective boundary layer (CBL) consists of three basic parts: (1) the surface layer unstably stratified and dominated by small-scale turbulence of very complex nature; (2) the CBL core dominated by the energy-, momentum-, and mass-transport of semi-organized structures (large-scale circulations), with a small contribution from small-scale turbulence produced by local structural shears; and (3) turbulent entrainment layer at the upper boundary, characterized by essentially stable stratification with negative (downward) turbulent flux of potential temperature. The energy- and flux budget theory developed previously for atmospheric stably-stratified turbulence and the surface layer in atmospheric convective turbulence is extended to the CBL core using budget equations for turbulent energies and turbulent fluxes of buoyancy and momentum. For the CBL core, we determine global turbulent characteristics (averaged over the entire volume of the semi-organized structure) as well as kinetic and thermal energies of the semi-organized structures as the functions of the aspect ratio of the semi-organized structure, the scale separation parameter between the vertical size of the structures and the integral scale of turbulence and the degree of thermal anisotropy characterized the form of plumes. The obtained theoretical relationships are potentially useful in modeling applications in the atmospheric convective boundary-layer and analysis of laboratory and field experiments, direct numerical simulations, and large-eddy simulations of convective turbulence with large-scale semi-organized structures.