Vertical Differencing for Quasi - Static Models

Vertical differencing is a very different problem from horizontal differencing. This may seem odd, but the explanation is very simple. There are three primary factors. First, gravitational effects are very powerful, and act only in the vertical. Second, the Earth's atmosphere is very shallow compared to its horizontal extent. Third, the atmosphere has a lower boundary. To construct a vertically discrete model, we have to make a lot of choices, including these: • The governing equations: Quasi-static or not? Shallow atmosphere or not? Anelastic or not? • The vertical coordinate system • The vertical staggering of the model's dependent variables • The properties of the exact equations that we want the discrete equations to mimic As usual, these choices will involve trade-offs. Each possible choice will have strengths and weaknesses. We must also be aware of possible interactions between the vertical differencing and the horizontal and temporal differencing. The speed of sound in the Earth's atmosphere is about 300 m s-1. If we permit vertically propagating sound waves, then, with explicit time differencing, the largest time step that is compatible with linear computational stability can be quite small. For example, if a model has a vertical grid spacing on the order of 300 m, the allowed time step will be on the order of 1 second. This may be palatable if the horizontal and vertical grid spacings are comparable. On the other hand, with a horizontal grid spacing of 30 km and a vertical grid spacing of 300 m, vertically propagating sound waves will limit the time step to about one percent of the value that would be compatible with the horizontal grid spacing. That's hard to take.