The appropriate upper boundary condition (UBC) formulation for the dynamical equations used in atmospheric physics is discussed in terms of both theoretical and computational aspects. The previous work on the UBC formulation is reviewed in the context of a linear mid-latitude primitive equation (PE) model. A new technique for constructing the UBC is introduced. The technique depends upon the existence of analytic solutions to simplifications to the equations of motion. These analytic solutions are used to construct the exact radiation UBCs (often used in tidal theory and studies of the upper atmosphere) which are non-local in time and space. Approximate UBCs, which are local in time, are formed through rational approximations to the exact radiation UBCs. The technique is demonstrated to be effective for both Rossby and gravity modes. The UBC is tested for computational problems initially in the linear PE model, and subsequently in a forced, damped nonlinear quasi-geostrophic model. 110, 1984-1993 Some problems in reproducing planetary waves by numerical models of the atmosphere. J. Met. SOC. Jap., 54, 129146 A simple boundary condition for unbounded hyperbolic flows. J . Comp. Phys., 21, 251-269 On the solution of the homogeneous vertical structure problem. J . Atmos. Sci., 36, 235C-2359 Quasi-geostrophic planetary wave forcing. Quart. J. R. Met. Atmospheric Tides. Aduances in Geoph., 7 , 105-187 Numerical forecasts of stratospheric warming events using a model with a hybrid vertical coordinate. Quart. J. R. Met. Approximation of pseudodifferential operators in absorbing boundary conditions for hyperbolic equations. Numer. Math., 42, 51-64 Oscillations of the earth's atmosphere. Cambridge University Press, New York and London Effect of viscosity on gravity waves and the upper boundary condition.
