Computation of the vertical velocity on the synoptic scale

Several sets of equations which can be used to find the vertical velocity are examined. A distinction is made between assumptions that are based on physical considerations and those based on computational necessity. Since the equations are solved as boundary value problems it is necessary to impose boundary conditions. These are discussed. Investigations are made into the use of the overrelaxation method for solving partial differential equations with either Dirichlet or Neumann boundary conditions. Emphasis is placed upon the determination of the optimum overrelaxation factor. A simple method of calculating this factor for the ω-equation is tested. The derivation, meaning and solution of the balance equation is discussed. New methods of solving this equation are introduced and are compared with existing methods. The boundary conditions for the linear balance equation are investigated and this leads to the derivation of a new boundary condition for the balance equation. The geostrophic ω-equation is examined and the elliptic condition is derived. Appropriate boundary conditions for ω are discussed and the effects of the form of the static stability on ω and ϴt are investigated. Simple models of the atmosphere are used from which several inferences are drawn. These are tested with case studies. The inconsistency of the usual boundary conditions for ω and ϴt, is also examined