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Numerical solutions to the equations
3.6 Upper-boundary conditions
 Artificial upper-boundary conditions are required in all atmospheric models because the model atmospheres do not extend to infinity. Indeed, for some historical applications the upper boundary has been located within the troposphere in order to save computational resources. An example of this approach is that Lavoie (1972) placed the upper model boundary, the “lid”, at the top of the boundary layer. Pielke (2002a) describes the location of the upper boundary in various historical model applications.
            Upward-propagating internal-gravity waves, for example generated by mountains or by deep convective storms, can extend to great heights in the atmosphere. Commonly used upper-boundary conditions (e.g., rigid lid, free surface) completely reflect these waves, which is a problem because no such reflection happens in nature, and erroneous downwardpropagating waves contaminate the model solution. There are a number of approaches for preventing this from happening. One involves the use in the model of a gravity-wave absorbing layer, or sponge layer, immediately below the model top, to prevent the wave from reaching the top and reflecting. Such wave absorption can be produced by employing a greatly enhanced, artificial horizontal and/or vertical diffusion (viscosity), where the viscosity increases from the standard value at the bottom of the layer to a maximum at the top boundary. A particular disadvantage of this approach is that the absorbing layer may need to be thick, spanning a large number of model layers and thus involving a large computational overhead. The overall effectiveness of the absorption depends on the wavelength of the gravity wave, the thickness of the absorbing layer, and the distribution of viscosity in the layer. Note that using a shallow absorbing layer with a very large, but computationally stable, viscosity will not be effective because large gradients in viscosity will also cause wave reflections. Klemp and Lilly (1978) defined the entire upper half of their computational domain as the absorbing layer. Figure 3.50 shows a two-dimensional model solution for idealized flow over a maximum in the orography, with and without the use of a viscous damping layer. The Gaussian obstacle had a 5-km half-width, and an amplitude of 1 km. Shown is the vertical motion field in the lowest 10 km of the 50-km-deep model. The model is described in Sharman and Wurtele (1983). The damping spanned the 20 km below a rigid lid that defined the model top. Without the damping, the reflected waves produce considerable noise in the troposphere, over 40 km below the model top. The waves in the solution for the experiment with the absorbing layer could be a result of imperfect damping, or more likely they could be a consequence of wave reflections from the lateral boundaries. An alternative approach for damping the waves before they reach the upper boundary is to use a Raleigh damping layer, again below the model top, where model variables are relaxed toward a predetermined reference state. For example, the Rayleigh damping term in a prognostic equation would be like


where α is any dependent variable, α is the reference value of that variable, and τ(z) increases upward within the damping layer and defines its vertical structure (e.g., see