1. Shapiro Filter A Shapiro Filter/smoother is used for time-dependent dynamical variables , , , , , , , and in the TIEGCM to limit the development of numerical nonlinear instability and short wavelength waves (less than four grid size) that are poorly represented in the model. Shapiro smoothing is used in modules/subroutines that solve equations for these variables. *Note: the following material is from Wang (1998), and copyright to the University of Michigan. Some of the material has been modified based on the updated TIEGCM. In real world, wave interactions occur that generate larger and smaller wavelength waves. The smaller waves cascade in sizes to reach the characteristic scale of molecular dissipation which is the process that finally eliminates motion. However, in a numerical model that has discrete grid, motion energy cascade from larger wavelength waves to smaller and smaller scales is interrupted. In fact, waves with wavelength smaller than 2?x (?x is the grid size of a numerical model) are erroneously represented as larger wavelength waves. This phenomenon is called aliasing (Pielke, 1984). The net result of this aliasing is a fictitious energy buildup since energy is added continuously to the model through forcing terms, while energy dissipation processes are cut off. Therefore, even though a numerical scheme is linearly stable, the results can degrade into physically meaningless computational noise. The solutions of model dynamical variables can grow to infinity. This numerical error is commonly referred to as nonlinear instability. To remove aliasing in the wave presentation, and thus to prevent the nonlinear instability from occurring, short waves with wavelength smaller than 4?x must be eliminated from a numerical model. In addition to the above mentioned nonlinear instability problem, such short waves are also poorly represented in terms of phase and amplitude. And they are expected to dissipate to even smaller scale motions anyway. Therefore, it is desirable to remove these waves completely. In the TIEGCM the smoothing technique discussed by Shapiro (1970) is used to control computational noise. The Shapiro smoother is a low-pass filter that eliminates waves with wavelength smaller than 4?x each time step, but leaves the large-scale disturbances unaffected. Another way to control nonlinear instability is to enhance the horizontal eddy diffusion by proper parameterization of the subgrid correlation terms. This method is not desirable since it allows modelers to adjust the eddy diffusion coefficient arbitrarily to change the magnitude of numerical solutions (Tag et al., 1979), A one-dimensional, five-point Shapiro smoothing scheme is used in both the TIEGCM to eliminate or constrain any spurious, nonlinear growth of high-frequency waves that may be introduced by roundoff and truncations errors or by the interruption of the energy cascade process that transport energy from large-scale disturbances to small-scale motions. Any time-dependent dynamical variable (, , , , , , , and ) is averaged each time step by (3.23) where I is the index of grid point in either latitude or longitude direction. is a smoothing factor and . can be expressed in terms of a summation of Fourier components of the form (3.24) where is a constant, A is the amplitude of the wave component with wave number (, where is the wavelength of the component). Substituting (3.24) into (3.23) yields (3.25) is set to be equal to 0.03. values are determined by numerical experiments. It is desirable to have as small as possible so that longer waves are less affected by the smoothing. The minimum value of is obtained by gradually decreasing while preserving integrity and stability of the numerical solutions. It is clear from (3.25) that all wavelengths are damped. However, short waves () are heavily damped while long waves are almost unaffected. Successive application of this smoother on a dynamical variable prevents short waves from growing to such a degree that they degrade the numerical solutions. 2. Fourier Filter *Note: the following material is from Wang (1998), and copyright to the University of Michigan. Some of the material has been modified based on the updated TIEGCM. Since the TIEGCM uses a uniform horizontal grid system and finite-difference numerical scheme, computational errors can grow significantly as the grids approach to the poles. The zonal grid sizes decrease in direct proportion to , where is the latitude. The truncation error of the finite-difference scheme and other numerical errors may be amplified by the zonal finite difference calculations at the grid points near poles. Non-physical waves are then generated and grow non-linearly to cause computational instability. This pole problem occurs in any simulation that uses finite difference techniques with uniform latitude and longitude grids. One way to avoid this problem is to use much smaller time steps, which is undesirable because it increases the computational cost dramatically by slowing down the entire simulation process. Another way to solve this pole problem is to apply a Fourier filter at high latitudes to remove non-physical high-frequency zonal waves generated by finite differences in the region with smaller zonal grid sizes. At each time step a Fourier expansion is applied to the prognostic variables of the model. A cutoff frequency was found at each latitude (at high latitudes) as a result of numerical experiments. Waves with frequencies that are higher than the cutoff frequencies are eliminated from the Fourier spectra of prognostic variables. The values of prognostic variables are then recovered through a reverse Fourier transform using modified Fourier spectra. Each latitude grid may have its own cutoff frequency. The cutoff frequency at a particular latitude is determined by the expected time step used by a model. A large time step requires low cutoff frequencies and more zonal waves are filtered out. In the TIEGCM, Fourier filters are applied to the individual prognostic variables. These variables include , , , , , , , and . It should be noted here that to apply a Fourier filter directly and successively on prognostic variables may cause overfiltering which may remove fine structures that are expected to occur at high latitudes. References: Pielke, R. A., Mesoscale Meteorological Modeling, Academic Press, Orlando, Florida, 1984. Shapiro, R., Smoothing, filtering, and Boundary effects, Rev. Geophys. Space Phys., 8, 359-387, 1970. Tag, P. M., Murry, F. W., and L. R. Kvenig, A comparison of several forms of eddy viscosity parameterization in a two-dimensional cloud model. J. Appl. Meteorol., 18, 1429-1441, 1979.