A multi-layer spectral model and the semi-implicit method

The formulaton of a multi-layer primitive equation model on the sphere is described. The horizontal representation is by means of spherical harmonics, truncated either in the triangular or rhomboidal manner. The time integration is performed using the semi-implicit method in which the linearized gravity wave terms are time averaged and thus the fast moving waves of this type are slowed. For a Slayer hemispheric model with triangular truncation at wavenumber 21, storage of 38K words is needed and with the time scheme allowing a time-step of 90 minutes, one day's simulation requires 11 seconds of CDC 7600 time. The growth of a baroclinic wave on a simple basic state of differential solid body rotation is exhibited. The errors involved in this case in utilizing the large time-step allowed by the semi-implicit scheme are thoroughly examined by comparing wave amplitudes and phases, conservation properties and gravity wave treatment for different time-steps. These errors are found to be negligible. The conservation properties of the model are in fact extremely good. The vertical finite differencing scheme of Arakawa is studied in the same baroclinic instability simulation. The growth is similar though the conservation of angular momentum is greatly improved. The transform method used in all these integrations allows some aliasing. but this is shown to be negligible.