In the realm of wave propagation through turbulent media, the spectrum of the orbital angular momentum of optical vortex beams is known to undergo symmetric broadening. However, the evolution of beams that are initially azimuthally asymmetric represents a distinct phenomenon. In this work, we have developed an analytical model describing the propagation of asymmetric OAM beams through the so-called Kolmogorov turbulence. Our results describe how the perturbation strength and the initial beam properties lead to a nonsymmetric spectrum of OAM modes. These findings lay the groundwork for further use of asymmetric fields that propagate in inhomogeneous media and their applications such as communications and sensing.