Semi-Implicit Formulations of the Navier--Stokes Equations: Application to Nonhydrostatic Atmospheric Modeling

We present semi-implicit (IMEX) formulations of the compressible Navier-Stokes equations (NSE) for applications in nonhydrostatic atmospheric modeling. The compressible NSE in nonhydrostatic atmospheric modeling include buoyancy terms that require special handling if one wishes to extract the Schur complement form of the linear implicit problem. We present results for five different forms of the compressible NSE and describe in detail how to formulate the semi-implicit time-integration method for these equations. Finally, we compare all five equations and compare the semi-implicit formulations of these equations both using the Schur and No Schur forms against an explicit Runge-Kutta method. Our simulations show that, if efficiency is the main criterion, it matters which form of the governing equations you choose. Furthermore, the semi-implicit formulations are faster than the explicit RungeKutta method for all the tests studied especially if the Schur form is used. While we have used the spectral element method for discretizing the spatial operators, the semi-implicit formulations that we derive are directly applicable to all other numerical methods. We show results for our five semi-implicit models for a variety of problems of interest in nonhydrostatic atmospheric modeling, including: inertia gravity waves, rising thermal bubbles (i.e., Rayleigh-Taylor instabilities), density current (i.e., Kelvin-Helmholtz instabilities), and mountain test cases; the latter test case requires the implementation of non-reflecting boundary conditions. Therefore, we show results for all five semi-implicit models using the appropriate boundary conditions required in nonhydrostatic atmospheric modeling: no-flux (reflecting) and non-reflecting boundary conditions. It is shown that the non-reflecting boundary conditions exert a strong impact on the accuracy and efficiency of the models.