The discrete formulation

The discrete formulation of the icosahedral hydrostatic dynamical core can be summarized as follows:

1. The spatial discretization

• The absolute vorticity flux, kinetic energy gradient, geopotencial gradient terms in the momentum equation and the continuity equation are discretized in essentially the same way as in the ICON shallow water model;

• The vertical advection of momentum and temperature are calculated in the same way as in ECHAM5 and in Simmons and Burridge (1981);

• The horizontal advection of temperature is discretized using the energy-conserving scheme
  
  in which div is the discrete divergence operator defined by Bonaventura and Ringler (2005);

• The hydrostatic equation is the same as in ECHAM5 and in Simmons and Burridge (1981). 

• The Simmons and Burridge (1981) discretization scheme for the pressure gradient force has been  adapted to the triangular C-grid. The resulting formula conserves the vertically integrated circulation along any closed curve that is composed of the dual edges of the horizontal grid.

• Formulation of the adiabatic heating is also adapted from Simmons and Burridge (1981).

　

2. The time integration

As in ECHAM5, the time integration scheme in the ICOHDC is based on the leap-frog scheme. A semi-implicit treatment for gravity waves is employed using a isothermal reference state. Growth of spurious computational modes is inhibited by the Asselin (1972) filter.