A series of high-resolution two- and three-dimensional model simulations of a severe thunderstorm are conducted to examine the sensitivity of above-cloud turbulence to a variety of background flow conditions, in particular the above-cloud wind shear and stability. Shortly after the initial convective overshoot, the above-cloud turbulence and mixing is dominated by mixing near the cloud boundary. At later times, when the convection is more mature, gravity wave breaking further aloft dominates the turbulence generation. This wave breaking is caused by critical level interactions, where the height of the critical level is controlled by the above-cloud wind shear. Furthermore, the strength of the above-cloud wind shear has a strong influence on the occurrence and intensity of above-cloud turbulence, with intermediate shears generating the largest volume of turbulent air, and strong shear conditions producing the most intense turbulence. Also, as expected, more stable situations are less prone to turbulence than less stable situations. These simulations will be presented, along with explanations of the mechanisms controlling the sensitivities, and a detailed comparison of the differences between the two- and three-dimensional results.