On the absolute instability of semi-implicit schemes for hydrostatic models

The dependence of the linear stability of two-time-level finite-difference semi-implicit schemes on the choice of reference temperature profile is studied. Particular vertical profiles of the temperature are considered to derive analytical conditions of stability. Analysis is made for general form of different model parameters such as the number of vertical levels and their distribution, the time step size, and the values of the viscosity coefficients. The derived conditions of stability are more restrictive than those for three-time-level schemes, but obtained necessary and sufficient condition for constant vertical lapse rates of the temperature has the form frequently applied to three-time-level schemes: the basic temperature profile should be warmer than the actual one. Performed numerical experiments show that the last restriction is neither necessary nor sufficient condition of stability for general temperature profiles.