If one takes a closed system of the basic meteorological equations and introduces within this system a finite expansion of the dependent variables using functions such as double Fourier or Fourier-Legendre functions in space, then the use of the orthogonality properties of these spatial functions enables one to obtain a set of coupled nonlinear ordinary differential equations for the coefficients of these functions. These coefficients are functions of time and the vertical coordinate, since the horizontal spatial dependence has been removed by taking a Fourier or a Fourier-Legendre transform of the equations. The coupled nonlinear ordinary differential equations for the coefficients are usually solved by simple time-differencing and vertical finite-differencing schemes. The mapping of the solution requires the multiplication of the coefficients with the spatial functions summed over a set of chosen finite spatial basis functions. This is what defines spectral modeling. Meteorological application of the spectral method was initiated by Silberman (1954), who studied the nondivergent barotropic vorticity equation in the spherical coordinate system using the spectral technique. In its earlier days, the spectral method was particularly suitable for low-resolution simple models. The equations of these simple models involved nonlinear terms evaluated at each time step. Evaluation of the nonlinear terms was performed using the interaction coefficient method and thus required large memory allocations, which was an undesirable proposition. However, with the introduction of the transform method, developed independently by Eliasen et al. (1970) and Orszag (1970), the method for evaluation of these nonlinear terms changed completely. This transform method also made it feasible to include nonadiabatic effects in the model equations. For the past couple of decades, the spectral method has be come an increasingly popular technique for studies of general circulation and numerical weather prediction at the operational and research centers. This method forms the basis for spectral modeling, and it is easy to understand if the reader has some background in linear algebra. We have a set of linearly independent functions θi(x), which are called the basis functions. The dependent variables of the problem are represented by a finite sum of these basis functions.
A Spectral Solenoidal-Galerkin Method for Rotating Thermal Convection between Rigid Plates
