Much of the theory and analysis for computations on the sphere is best
understood in the context of Fourier theory and analysis in Cartesian
geometry. In addition, discrete harmonic analysis begins with Fourier
analysis in the longitudinal direction. Therefore we begin our study
of computations on the sphere with a review of topics in discrete
Fourier analysis. Click here to view the notes.
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| Trig. representations
| Spectral accuracy
| Nonperiodic functions
| The discrete basis
| Aliasing
| Trig interpolation
| Interpolation error
| Alias control
| Two-thirds rule
| Subroutine EZFFT
| Using EZFFT
| FFT for any N
| Staggered grids
| Complex transform
| Real in terms of complex
| The FFT
| Multiprocessor FFTs
| Symmetric FFTs
| FFTPACK
| Accessing FFTPACK
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Here we discuss the basic tools that are used for the spectral
representation of scalar functions
(such as temperature, pressure, divergence) on the sphere.
Click here to view notes.
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| Sphere vs rectangle
| Least squares representation
| Assoc. Legendre functions
| Double Fourier series
| Computing the ALFs
| Integration formulas
| ALFPACK
| Gauss pts & weights
| Scalar harmonic analysis
| Generalized harmonic analysis
| Selecting the finite basis
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Vectors on the sphere are discontinuous (multivalued) at the poles and
therefore scalar spectral analysis cannot be applied to the individual
components like Fourier analysis of vectors on the rectangle
Important examples for geophysical applications include the
wind and magnetic field. Click here to view
notes.
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| Discontinuous vectors
| Vector harmonic analysis
| Unbounded derivatives
| Computing vorticity
| Bounded differential
| divergence and gradients
| expressions
| Robert's U, V variables
| Vector harmonics
| and attributes
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Here we describe the spectral transform method for modeling geophysical
fluids. Actually we discuss two popular methods plus the vector harmonic
transform method and note that others exist. The methods are presented by
application to the shallow water equations.
Click here to view notes.
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| Shallow water equations (SWE)
| SWE with bounded terms
| Ritchie's U, V model (ECMWF)
| Vorticity & divergence formulation (NCAR)
| Vector harmonic method & attributes
| Model results
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