Trying to achieve the no flow condition in MC2

In MC2, a semi-implicit semi-Lagrangian scheme is used to solve the Euler equations of motion in height-base terrain-following coordinates. Achieving the no flow condition, i.e. a perfect state of rest, is a problem with this approach since the hydrostatic equation is then a non-linear three-dimensional diagnostic equation with a discretized form involving temperature and pressure deviations at three time levels. In fact, exact equilibrium can only be achieved if the deviations vanish, i.e. when the true state of the model and its basic state coincide. In 2002, we have eliminated the previous restriction to isothermal basic state to allow for any basic state. Nevertheless, as soon as we considered model states different from its basic state, there occurred oscillations. In absence of off-centering, the amplitude of the oscillations depended on the distance between the two states and for large enough differences the model would become unstable, even in the absence of topography. Since then, we have found a modification to the semi-implicit scheme that eliminates oscillations and therefore stabilizes the scheme. We have also found a way to make the hydrostatic equation linear (no approximation implied) with therefore the possibility to apply one all time levels separately. We explain these changes and show the results.