Stability of atmospheric gravity waves
Final Report Abstract
Stratified fluids such as the Earth’s atmosphere give rise to gravity waves. The majority of these waves are excited in the lowest 10 km where they are effectively described by linear theory since their amplitudes are small. The waves grow in amplitude as they propagate upwards due to the decreasing background density. Some of them reach far into the “deep” atmosphere where their amplitudes cannot be considered small anymore. Here, nonlinear wave theory governs the dynamics. State of the art numerical weather forecasting models project the state variables, such as temperature, wind velocity, pressure etc., onto a grid covering the entire globe with typical horizontal spacing of 10 km. Gravity waves, however, have occasionally wavelengths much smaller than this resolution threshold which means that they cannot be “seen” by the model. Nevertheless, these waves have a significant impact on the weather and must not be ignored. A circumnavigation around this issue are parametrizations which model such unresolved processes in terms of the resolved state variables. Most uptodate parametrizations rely on linear theory. This project aims to increase the accuracy of weather forecasting by extending the wave theory including nonlinear effects. One of which is wave instability. The governing equations for our theoretical analysis are modulation equations that result from nonlinear Wentzel-Kramers-Brillouin asymptotic theory of the fundamental system: the compressible Navier-Stokes equations. We discovered that the modulation equations exhibit a certain kind of solution, traveling waves, which are analytically well understood. Our solutions possess in particular counterintuitive properties that are certainly not available in linear theory. For instance, the group velocity, as defined usually by the derivative of the dispersion relation, may not relate to the wave’s actual envelope velocity. Exploiting advanced analytical as well as numerical methods of applied mathematics like spectral stability analysis and total variation diminishing finite-volume methods, we found out that these waves are unconditionally unstable. Those results have direct implications for the parametrization algorithms.
Publications
- (2018), Spectral stability of nonlinear gravity waves in the atmosphere
Schlutow, M., Wahlén, E. and Birken, P.
(See online at https://doi.org/10.1515/mcwf-2019-0002)
