Preliminary results from the application of our recently developed extension of the Dynamic Smagorinsky model (DSM) are presented. The DSM serves as a parameterization for subgrid-scale momentum diffusion in a general circulation model (GCM) that allows for stable simulations even without a hyperdiffusion. Due to gravity, the horizontal and vertical scales have to be treated differently. Our extension includes a test filter for the Smagorinsky parameter that is separated from the resolution scale, and a dynamic approach for the vertical diffusion based on the ideas of stratified turbulence. The aim of our investigations is to extend our model into the MLT for a study of GW spectra with respect to the scaling ratio of stratified macro-turbulence. Therefore, we show the performance of the DSM up to the lower stratosphere and discuss the implications from the scale-invariance criterion for LES parameterizations on the approach for dynamic vertical diffusion.