The fluctuation characteristics of the orbital angular momentum (OAM) are studied numerically in the computer experiment simulating the propagation of an OAM-carrying laser beam through a turbulent atmosphere and OAM registering by a finite-size receiver aperture. The wave-optics numerical simulation employs the conventional split-operator method (with phase screens) and the Monte Carlo averaging technique. OAM statistical characteristics in the turbulent medium are compared for two employed definitions of the specific (normalized) orbital angular momentum. For the non-vortex Gaussian beam, the correlation between OAM fluctuations and intercepted power is found to be negligibly small at arbitrary size of the receiver aperture. For the vortex Laguerre—Gaussian beam, the correlation between OAM fluctuations and intercepted power is significant (the correlation coefficient can achieve 0.7). In this case, the values of OAM variance for these definitions differ almost twice. It is shown that the OAM value averaged over realizations of the turbulent medium for the aperture size intercepting almost the entire beam power becomes equal to that in the homogeneous medium. As the aperture size decreases, the average OAM value decreases, and when the aperture size tends to zero, it tends to zero as well. The results obtained in the paper can be used in development and improvement of methods for optical transmission of information by optical beams with OAM, as well as in development of optical means for manipulating micron-sized particles.