The reliability of a vertical underwater wireless optical communication (UWOC) network is seriously impacted by turbulence-induced fading due to fluctuations in the water temperature and salinity, which vary with depth. To better assess the vertical UWOC system performances, an accurate probability distribution function (PDF) model that can describe this fading is indispensable. In view of the limitations of theoretical and experimental studies, this paper is the first to establish a more accurate modeling scheme for wave optics simulation (WOS) by fully considering the constraints of sampling conditions on multi-phase screen parameters. On this basis, we complete the modeling of light propagation in a vertical oceanic turbulence channel and subsequently propose a unified statistical model named mixture Weibull-generalized Gamma (WGG) distribution model to characterize turbulence-induced fading in vertical links. Interestingly, the WGG model is shown to provide a perfect fit with the acquired data under all considered channel conditions. We further show that the application of the WGG model leads to closed-form and analytically tractable expressions for key UWOC system performance metrics such as the average bit-error rate (BER). The presented results give valuable insight into the practical aspects of development of UWOC networks.