Search Results
Abstract
Modified Fourier series, as judged by criteria of accuracy, numerical efficiency and ease of programming, are the best choice of latitudinal expansion functions for general problems on the sphere. The pseudospectral and spectral methods, however, can be easily and successfully applied with all three types of orthogonal series. For special situations, such as when the latitude variable is stretched, Chebyshev polynomials are the only practical choice, but for orthodox problems on the globe, they are less efficient than the other two sets of functions. Although spherical harmonics have been universally employed in the past, Fourier series give comparable accuracy and are significantly easier to program and manipulate. Thus, in the absence of a special reason to the contrary, the simplest and most effective way to handle the north–south dependence of the solution to a boundary or eigenvalue problem on the sphere is to use a Fourier series in colatitude.
Abstract
Modified Fourier series, as judged by criteria of accuracy, numerical efficiency and ease of programming, are the best choice of latitudinal expansion functions for general problems on the sphere. The pseudospectral and spectral methods, however, can be easily and successfully applied with all three types of orthogonal series. For special situations, such as when the latitude variable is stretched, Chebyshev polynomials are the only practical choice, but for orthodox problems on the globe, they are less efficient than the other two sets of functions. Although spherical harmonics have been universally employed in the past, Fourier series give comparable accuracy and are significantly easier to program and manipulate. Thus, in the absence of a special reason to the contrary, the simplest and most effective way to handle the north–south dependence of the solution to a boundary or eigenvalue problem on the sphere is to use a Fourier series in colatitude.
Abstract
It is shown that spectral and pseudospectral methods can lead to great savings in numerical efficiency-typically, at least a factor of four in storage and eight in operation count-in comparison with finite-difference methods for solving eigenvalue and nonseparable, noniterative boundary value problems at the cost of only a a slight amount of additional programming. The special problems or apparent problems that geophysical boundary layers, critical latitudes and critical levels and real data create for spectral algorithms are also discussed along with when and how these can be solved.
Abstract
It is shown that spectral and pseudospectral methods can lead to great savings in numerical efficiency-typically, at least a factor of four in storage and eight in operation count-in comparison with finite-difference methods for solving eigenvalue and nonseparable, noniterative boundary value problems at the cost of only a a slight amount of additional programming. The special problems or apparent problems that geophysical boundary layers, critical latitudes and critical levels and real data create for spectral algorithms are also discussed along with when and how these can be solved.
Abstract
Regional spectral models have previously periodized and blended limited-area data through ad hoc low-order schemes justified by intuition and empiricism. By using infinitely differentiable “window functions” or “bells” borrowed from wavelet theory, one can periodize with preservation of spectral accuracy. Similarly, it is shown through a mixture of theory and numerical examples that Davies relaxation for blending limited-area and global data in one-way nested forecasting can be performed using the same C ∞ bells as employed for the Fourier blending.
“The relative success of empirical methods . . . may be used as partial justification to allow us to make the daring approximation that the data on a limited area domain may be decomposed into a trend and a periodic perturbation, and to proceed with Fourier transformation of the latter.” Laprise (2003, p. 775)
Abstract
Regional spectral models have previously periodized and blended limited-area data through ad hoc low-order schemes justified by intuition and empiricism. By using infinitely differentiable “window functions” or “bells” borrowed from wavelet theory, one can periodize with preservation of spectral accuracy. Similarly, it is shown through a mixture of theory and numerical examples that Davies relaxation for blending limited-area and global data in one-way nested forecasting can be performed using the same C ∞ bells as employed for the Fourier blending.
“The relative success of empirical methods . . . may be used as partial justification to allow us to make the daring approximation that the data on a limited area domain may be decomposed into a trend and a periodic perturbation, and to proceed with Fourier transformation of the latter.” Laprise (2003, p. 775)
Abstract
We introduce a new energetics concept and apply it to the NCAR Community Climate Model. The new features of our approach are that the energy is split into balanced and transient parts and that the balanced energy consists of rotational energy and balanced gravitational energy. The time evolution and distribution of the balanced and transient parts of the gravity waves among vertical modes and zonal waves am analyzed.
Both balanced gravitational energy and transient energy concentrate and oscillate rapidly with time at vertical modes 7–8 and zonal wavenumbers 1–5. This explains why the iteration scheme used in nonlinear normal mode initialization would not converge, in general, for high vertical modes and long zonal waves. All the gravity waves associated with vertical modes 0–2 and any zonal wavenumber can be freely adjusted in the initialization to suppress the high-frequency oscillations.
The lower vertical modes, 2–6, contain more balanced gravitational energy than transient energy, but for the higher baroclinic modes, 7–8, both energies are of almost the same magnitude. In general, longer zonal waves contribute more energy to the balanced gravitational energy. Zonal wavenumber 1 contributes the most to both transient energy and balanced gravitational energy. To examine whether the energy of gravity waves is balanced or not during the initialization, it is inappropriate to express the energy in terms of zonal wavenumbers only. The vertical resolution, discretization scheme, and physical parameterization may distort the gravitational energy in the high vertical modes.
Abstract
We introduce a new energetics concept and apply it to the NCAR Community Climate Model. The new features of our approach are that the energy is split into balanced and transient parts and that the balanced energy consists of rotational energy and balanced gravitational energy. The time evolution and distribution of the balanced and transient parts of the gravity waves among vertical modes and zonal waves am analyzed.
Both balanced gravitational energy and transient energy concentrate and oscillate rapidly with time at vertical modes 7–8 and zonal wavenumbers 1–5. This explains why the iteration scheme used in nonlinear normal mode initialization would not converge, in general, for high vertical modes and long zonal waves. All the gravity waves associated with vertical modes 0–2 and any zonal wavenumber can be freely adjusted in the initialization to suppress the high-frequency oscillations.
The lower vertical modes, 2–6, contain more balanced gravitational energy than transient energy, but for the higher baroclinic modes, 7–8, both energies are of almost the same magnitude. In general, longer zonal waves contribute more energy to the balanced gravitational energy. Zonal wavenumber 1 contributes the most to both transient energy and balanced gravitational energy. To examine whether the energy of gravity waves is balanced or not during the initialization, it is inappropriate to express the energy in terms of zonal wavenumbers only. The vertical resolution, discretization scheme, and physical parameterization may distort the gravitational energy in the high vertical modes.
Abstract
A new approach to energetics is introduced and applied to the NCAR Community Climate Model. All the energy components are separated into gravitational and rotational parts. The new feature of our scheme is that the gravitational divergent kinetic energy is further decoupled into zonal and meridional components, which measure the strength of the cut-west and meridional circulations, respectively. The zonal and meridional nondivergent kinetic energies represent the vorticities related to the nondivergent zonal and meridional winds, respectively. The distributions of energy among meridional indices, vertical modes, and zonal waves are analyzed.
We suggest a new, easy, and reasonable criterion to adjust the gravity waves in the initialization based on the vertical modes, meridional indices, and zonal wavenumbers. To retain the strength of large-scale circulations and to preserve the intensity of synoptic scale pressure systems, we recommend that the gravity waves corresponding to the internal modes 2–6, meridional indices 1–6, and zonal wavenumbers 1–10 not be adjusted significantly during the initialization. However, all the other gravity waves can be adjusted initially, particularly those associated with the external and the first internal modes.
Abstract
A new approach to energetics is introduced and applied to the NCAR Community Climate Model. All the energy components are separated into gravitational and rotational parts. The new feature of our scheme is that the gravitational divergent kinetic energy is further decoupled into zonal and meridional components, which measure the strength of the cut-west and meridional circulations, respectively. The zonal and meridional nondivergent kinetic energies represent the vorticities related to the nondivergent zonal and meridional winds, respectively. The distributions of energy among meridional indices, vertical modes, and zonal waves are analyzed.
We suggest a new, easy, and reasonable criterion to adjust the gravity waves in the initialization based on the vertical modes, meridional indices, and zonal wavenumbers. To retain the strength of large-scale circulations and to preserve the intensity of synoptic scale pressure systems, we recommend that the gravity waves corresponding to the internal modes 2–6, meridional indices 1–6, and zonal wavenumbers 1–10 not be adjusted significantly during the initialization. However, all the other gravity waves can be adjusted initially, particularly those associated with the external and the first internal modes.
Abstract
Bivariate Fourier series have many benefits in limited-area modeling (LAM), weather forecasting, and meteorological data analysis. However, atmospheric data are not spatially periodic on the LAM domain (“window”), which can be normalized to the unit square (x, y) ∈ [0, 1] ⊗ [0, 1] by rescaling the coordinates. Most Fourier LAM meteorology has employed rather low-order methods that have been quite successful in spite of Gibbs phenomenon at the boundaries of the artificial periodicity window. In this article, the authors explain why. Because data near the boundary between the high-resolution LAM window and the low-resolution global model are necessarily suspect, corrupted by the discontinuity in resolution, meteorologists routinely ignore LAM results in a buffer strip of nondimensional width D, and analyze only the Fourier sums in the smaller domain (x, y) ∈ [D, 1 − D] ⊗ [D, 1 − D]. It is shown that the error in a one-dimensional Fourier series with N terms or in a two-dimensional series with N 2 terms, is smaller by a factor of N on a boundary-buffer-discarded domain than on the full unit square. A variety of procedures for raising the order of Fourier series convergence are described, and it is explained how the deletion of the boundary strip greatly simplifies and improves these enhancements. The prime exemplar is solving the Poisson equation with homogeneous boundary conditions by sine series, but the authors also discuss the Laplace equation with inhomogeneous boundary conditions.
Abstract
Bivariate Fourier series have many benefits in limited-area modeling (LAM), weather forecasting, and meteorological data analysis. However, atmospheric data are not spatially periodic on the LAM domain (“window”), which can be normalized to the unit square (x, y) ∈ [0, 1] ⊗ [0, 1] by rescaling the coordinates. Most Fourier LAM meteorology has employed rather low-order methods that have been quite successful in spite of Gibbs phenomenon at the boundaries of the artificial periodicity window. In this article, the authors explain why. Because data near the boundary between the high-resolution LAM window and the low-resolution global model are necessarily suspect, corrupted by the discontinuity in resolution, meteorologists routinely ignore LAM results in a buffer strip of nondimensional width D, and analyze only the Fourier sums in the smaller domain (x, y) ∈ [D, 1 − D] ⊗ [D, 1 − D]. It is shown that the error in a one-dimensional Fourier series with N terms or in a two-dimensional series with N 2 terms, is smaller by a factor of N on a boundary-buffer-discarded domain than on the full unit square. A variety of procedures for raising the order of Fourier series convergence are described, and it is explained how the deletion of the boundary strip greatly simplifies and improves these enhancements. The prime exemplar is solving the Poisson equation with homogeneous boundary conditions by sine series, but the authors also discuss the Laplace equation with inhomogeneous boundary conditions.
Abstract
Accurate and stable numerical discretization of the equations for the nonhydrostatic atmosphere is required, for example, to resolve interactions between clouds and aerosols in the atmosphere. Here the authors present a modification of the hydrostatic control-volume approach for solving the nonhydrostatic Euler equations with a Lagrangian vertical coordinate. A scheme with low numerical diffusion is achieved by introducing a low Mach number approximate Riemann solver (LMARS) for atmospheric flows. LMARS is a flexible way to ensure stability for finite-volume numerical schemes in both Eulerian and vertical Lagrangian configurations. This new approach is validated on test cases using a 2D (x–z) configuration.
Abstract
Accurate and stable numerical discretization of the equations for the nonhydrostatic atmosphere is required, for example, to resolve interactions between clouds and aerosols in the atmosphere. Here the authors present a modification of the hydrostatic control-volume approach for solving the nonhydrostatic Euler equations with a Lagrangian vertical coordinate. A scheme with low numerical diffusion is achieved by introducing a low Mach number approximate Riemann solver (LMARS) for atmospheric flows. LMARS is a flexible way to ensure stability for finite-volume numerical schemes in both Eulerian and vertical Lagrangian configurations. This new approach is validated on test cases using a 2D (x–z) configuration.
