Employing the theory of coherence, the partially coherent flat-topped vortex hollow beam is first introduced. The analytical propagation equation for a partially coherent flat-topped vortex hollow beam in oceanic turbulence is derived by using the extended Huygens–Fresnel diffraction integral and the spectrum of oceanic turbulence. The influences of coherence length, beam order $N$, the topological charge $M$, and the parameters of oceanic turbulence on evolution properties of partially coherent flat-topped vortex hollow beams propagating in oceanic turbulence are obtained by using the numerical examples.

The partially coherent Lorentz–Gauss vortex beam generated by a Schell-model source has been introduced. Based on the extended Huygens–Fresnel principle, the cross-spectral density function of a partially coherent Lorentz–Gauss vortex beam propagating in oceanic turbulence is derived. The influences of coherence length, topological charge M, and oceanic turbulence on the spreading properties and position of the coherence vortex for a partially coherent Lorentz–Gauss vortex beam are analyzed in detail. The results show that a partially coherent Lorentz–Gauss vortex beam propagating in stronger oceanic turbulence will evolve into a Gaussian-like beam more rapidly as the propagation distance increases, and the number of coherent vortices will change.

Based on the extended Huygens–Fresnel principle, the analytical expressions for partially coherent four-petal Gaussian vortex beams propagating in oceanic turbulence are obtained, and the influence of the coherence length, beam order N, topological charge M and oceanic turbulence parameters on the evolution properties of beams are discussed in detail using numerical examples. The results show that the beam will evolve into a Gauss-like beam rapidly with decreasing coherence length and oceanic parameters and , or increasing oceanic parameter in the far field due to the influence of the coherence length and oceanic turbulence.

The analytical propagation equations for partially coherent four-petal Gaussian vortex beams propagating through uniaxial crystals orthogonal to the optical axis are derived, and the propagation properties are analyzed by numerical examples. The influence of beam order $n$, topological charge M, the coherence length, and the ratio of refractive index $n_e/n_o$ on the normalized intensity distribution of partially coherent four-petal Gaussian vortex beams is discussed.