Figure 19. (a) Horizontal plane of numerical simulation and (b) vertical profiles of analytic solution (dashed line) and numerical result (solid line) of Doswell's idealized cyclogenesis after 16 time steps with courant number = 4.243, revolution = 4.386 and S (transition width) = 2.0 with mass correction but without internet filter (Sun and Sun, 2004).

Oh (2007) applied the semi-Lagrangian integration method (Sun et al., 1996; Sun and Yeh, 1997) combined with the method of characteristics to the 2D shallow water equations to study the geostrophic adjustment problem, the shear instability, the collapse of a circular dam, the interaction of a vortex with the terrain, and the merging of vortices. His results show that the scheme is highly accurate in simulating flows involving a sharp gradient, such as the collapse of the cylindrical dam and the merging of two vortices. The characteristic approach is easier to interpret. Figure 20(a) shows the height at different times, and Fig. 20(b) the PV (potential vorticity), revealing that the height (or kinetic energy) cascades to the larger wavelength while the vorticity field cascades to the smaller wavelength in a 2D flow.

For the past several decades, numerical models have substantially improved in simulating short-term weather and long-term climate change. However, tremendous work is still required to improve the basic equations, numerical methods, resolvable and subgrid scale physical parametrizations, and initial and boundary conditions (Sun, 2002). More observational data is also required to provide better initial and boundary conditions. The reanalysis data, which has been used for initial and boundary conditions in weather/climate simulations, is a combination of model output and observations. More observations are also needed to validate the model results. Finally, we need better computational resources (both hardware and software) in order to develop new models with finer resolution and comprehensive

physics/parametrizations, as well as to display model results.

Numerical modeling is an exciting and challenging field. With the appropriate equations, numerical methods, and initial and boundary conditions, the models can reveal the spatial and temporal evolution of the processes involved. Models can also be used to simulate climate/environment in the past or in the future, which cannot be carried out in the laboratory or in field experiments. Hence, numerical models are the most important tool in weather forecasting and climate study. However, we should also be aware that uncertainty

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