SpectralSphericalMesh

Spherical mesh data structure for calculating spectra. The mesh represents a regular lat-long grid.

power_spectrum_1d(FT, var_grid, z, lat, lon, weight)

For a variable var_grid on a (lon,lat,z) grid, given an array of weights, compute the zonal (1D) power spectrum using a Fourier transform at each Gaussian latitude. The input field must be first intepolated to a Gaussian grid.

Arguments

• FT: FloatType
• var_grid: variable on a Gaussian (lon, lat, z) grid to be transformed
• z: Array with uniform z levels
• lat: Array with uniform lats
• lon: Array with uniform longs
• weight: Array with weights for mass-weighted calculations

power_spectrum_2d(FT, var_grid, mass_weight)

Transform a variable defined on a regular lat long grid to the 2d spectral space using fft on latitude circles (as for the 1D spectrum) and Legendre polynomials for meridians, and calculate spectra.

Arguments

• FT: FloatType
• var_grid: variable on a Gaussian (lon, lat, z) grid to be transformed
• mass_weight: Array with weights for mass-weighted calculations.

References

• [21]

compute_gaussian!(FT, n)

Compute sin(latitude) and the weight factors for Gaussian integration.

Arguments

• FT: FloatType
• n: Int, number of Gaussian latitudes

References

• Ehrendorfer, M., Spectral Numerical Weather Prediction Models, Appendix B, Society for Industrial and Applied Mathematics, 2011

Details (following notation from Ehrendorfer, 2011):

Pn(x) is an odd function
solve half of the n roots and weightes of Pn(x) # n = 2n_half
P_{-1}(x) = 0
P_0(x) = 1
P_1(x) = x
nP_n(x) = (2n-1)xP_{n-1}(x) - (n-1)P_{n-2}(x)
P'_n(x) = n/(x^2-1)(xP_{n}(x) - P_{n-1}(x))
x -= P_n(x)/P'_{n}()
Initial guess xi^{0} = cos(π(i-0.25)/(n+0.5))
wi = 2/(1-xi^2)/P_n'(xi)^2

compute_legendre!(FT, num_fourier, num_spherical, sinθ, nθ)

Normalized associated Legendre polynomials, P_{m,l} = qnm.

Arguments:

• FT: FloatType
• num_fourier: Int, number of truncated zonal wavenumbers (m)
• num_spherical: Int, number of total wavenumbers (n)
• sinθ: Array{FT} with sin(latitude)
• nθ: Int, number of Gaussian latitudes

References:

• Ehrendorfer, M. (2011) Spectral Numerical Weather Prediction Models, Appendix B, Society for Industrial and Applied Mathematics
• Winch, D. (2007) Spherical harmonics, in Encyclopedia of Geomagnetism and Paleomagnetism, Eds Gubbins D. and Herrero-Bervera, E., Springer

Details (using notation and Eq. references from Ehrendorfer, 2011):

l=0,1...∞    and m = -l, -l+1, ... l-1, l
P_{0,0} = 1, such that 1/4π ∫∫YYdS = δ (where Y = spherical harmonics, S = domain surface area)
P_{m,m} = sqrt((2m+1)/2m) cosθ P_{m-1,m-1}
P_{m+1,m} = sqrt(2m+3) sinθ P_{m,m}
sqrt((l^2-m^2)/(4l^2-1))P_{l,m} = P_{l-1, m} -  sqrt(((l-1)^2-m^2)/(4(l-1)^2 - 1))P_{l-2,m}
THe normalization assures that 1/2 ∫_{-1}^1 P_{l,m}(sinθ) P_{n,m}(sinθ) dsinθ = δ_{n,l}
Julia index starts with 1, so qnm[m+1,l+1] = P_l^m

trans_grid_to_spherical!(mesh::SpectralSphericalMesh, pfield::Arr{FT,2})

Transforms a variable on a Gaussian grid (pfield[nλ, nθ]) into the spherical harmonics domain (varspherical2d[numfourier+1, num_spherical+1]).

Details:

Here λ = longitude, θ = latitude, η = sinθ, m = zonal wavenumber, n = total wavenumber:
var_spherical2d = F_{m,n}    # Output variable in spectral space (Complex{FT}[num_fourier+1, num_spherical+1])
qwg = P_{m,n}(η)w(η)         # Weighted Legendre polynomials (FT[num_fourier+1, num_spherical+1, nθ])
var_fourier2d = g_{m, θ}     # Untruncated Fourier transformation (Complex{FT} [nλ, nθ])
pfield = F(λ, η)             # Input variable on Gaussian grid FT[nλ, nθ]

Arguments

• mesh: struct with mesh information
• pfield: variable on Gaussian grid to be transformed

References

• Ehrendorfer, M., Spectral Numerical Weather Prediction Models, Appendix B, Society for Industrial and Applied Mathematics, 2011
• [22]

compute_wave_numbers!(wave_numbers, num_fourier::Int, num_spherical::Int)

Store the total wave number n for this basis in a matrix wave_numbers of shape [m,n].

Arguments:

• wavenumbers: Matrix of [Int, Int] to store the wave wavenumbers
• num_fourier: Int, number of truncated zonal wavenumbers (m)
• num_spherical: Int, number of total wavenumbers (n)