The Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST) will image tens of billions of galaxies over ten years–a 100 times increase over previous surveys–enabling the most precise measurements of dark energy properties to date. Realizing the full potential of this powerful dataset requires an unprecedented understanding of observational effects that, if uncorrected, can bias the science results. One such source is the blurring of images due to optical turbulence in the atmosphere, which dominates the point-spread function (PSF) for ground-based instruments. This atmospheric effect imprints spatially correlated noise on scales (and with amplitudes) similar to the cosmological signal we will study: the spatial correlation of positions and shapes of galaxies on the plane of the sky due to gravitational lensing of light by the dark matter in the Universe. High-fidelity simulated astronomical images are an important tool in developing and measuring the performance of image-processing algorithms that will be needed to accurately and precisely account for sources of correlated noise such as the atmospheric PSF. This thesis presents a new simulation tool, psf-weather-station, that allows us to study and model the dependence of correlations in the atmospheric PSF on weather conditions at any observatory by leveraging data from weather forecasting models. We use this tool to simulate and study the correlations in the size and shape of the atmospheric PSF predicted for the Vera C. Rubin Observatory in Chile. We make quantitative predictions for two-point correlation functions (2PCF) that are used in analyses of cosmic shear. We observe a strong anisotropy in the two-dimensional 2PCF, which is expected based on observations in real images, and study the dependence of the orientation of the anisotropy on dominant wind directions at the observatory site. We also explore the temporal behaviour of the atmospheric PSF using high-resolution speckle images of stars, recorded near Rubin Observatory, and simulate similar observations using psf- weather-station. We find differences in the time scales of correlation in the PSF between data and simulations, but observe a consistent dependence of PSF shape with image motion on short time scales.