Spectral fields

References: [Davies, 1990]

The ECMWF definition of a spherical harmonic function is the following:

$$X(\lambda,\mu)  =  \sum\limits_{m=-M}^M \underbrace{\su... ...=\vert m\vert}^N X_n^m P_n^m(\mu)}_{X^m(\mu)} e^{im\lambda}$$ (3.2)

where the associated Legendre functions are given by

$$P_n^m(\mu) = \sqrt{(2n+1)\frac{(n-m)!}{(n+m)!}} \frac{1}{2^nn!}\ (1-\mu^2)^{m/2} \frac{\mbox{d}^{n+m}}{\mbox{d}\mu^{n+m}} (\mu^2-1)^n$$ (3.3)

$$P_n^{-m}(\mu) = P_n^m(\mu)$$ (3.4)

Given $P_{m-1}^m(\mu)=0$ and $P_m^m(\mu)$, a stable recurrent relation is available for $P_n^m(\mu)$. The derivative of Legendre function is often required:

$$(1-\mu^2)\frac{\mbox{d}P_n^m}{\mbox{d}\mu} = (n+1) \epsilon_n^m P_{n-1}^m -  n \epsilon_{n+1}^m P_{n+1}^m \qquad,\qquad \epsilon_n^m =  \sqrt{\frac{n^2-m^2}{4n^2-1}}$$ (3.5)

For a real variable $X(\lambda ,\mu )$, the sphericial coefficients $X_n^m$ satisfy:

$$X_n^{-m} =  \overline{X_n^m}$$ (3.6)

Eigenfunctions of Laplacian

The sperical harmonics are eigenfunction of the Laplacian operator on the sphere:

$$\nabla^2\{P_n^m(\mu) e^{im\lambda}\} =  -\frac{n(n+1)}{r^2} P_n^m(\mu) e^{im\lambda}$$ (3.7)

Fourrier coefficients

The "Fourrier coefficients" $X^m(\mu)$ are usefull for evaluation of the spherical harmonics:

$$X^m(\mu) =  \sum\limits_{n=\vert m\vert}^N X_n^m P_n^m(\mu)$$ (3.8)

For a real variable $X(\lambda ,\mu )$, the Fourrier coefficients $X^m$ satisfy:

$$X^{-m}(\mu) =  \overline{X^m(\mu)}$$ (3.9)

Evaluation over multiple longitudes

Use Fast Fourrier Transform to evaluate at $J\ge 2M+1$ equiradial spaced longitudes including zero:

$$\lambda_j = j 2\pi/J \quad,\qquad j=0,\dots,J-1$$ (3.10)

$$X^m(\mu) = 0 \quad,\qquad m=M+1,\dots, J/2 \qquad\mbox{(truncated)}$$ (3.11)

$$X(\lambda_j,\mu) = \sum\limits_{m=-M}^M X^m(\mu) e^{im\lambda_j}$$ (3.12)

$$= \sum\limits_{m=0}^M X^m(\mu) e^{im\lambda_j}  +  \sum\limits_{m=-M}^{-1} X^m(\mu) e^{im\lambda_j}$$ (3.13)

$$= \sum\limits_{m=0}^{J/2} X^m(\mu) e^{im\lambda_j}  +  \sum\limits_{m=J/2+1}^{J-1} X^{m-J}(\mu) e^{i(m-J)\lambda_j}$$ (3.14)

$$= \sum\limits_{m=0}^{J/2} X^m(\mu) e^{im\lambda_j}  +  \sum\limits_{m=J/2+1}^{J-1} \overline{X^{2J-m}(\mu)} e^{im\lambda_j}$$ (3.15)

$$= \sum\limits_{m=0}^{J-1} C^m(\mu) e^{im\lambda_j} \qquad C^m(\mu)... ...l} X^m(\mu) &,& m < J/2 \\ \\ \overline{X^{J-m}} &,& m>J/2 \end{array}\right.$$ (3.16)

$$= \mbox{FFT99}( J, \underbrace{X^0(\mu),\dots,X^M(\mu),0,\dots,0}_{1+J/2 \mbox{complex numbers}} )$$ (3.17)

For implementation with emos routine fft99, choose $J$ compsed from factors 2, 3, and 5 .

Evaluation at oposite latitudes

The Fourrier coefficients $X^m(\mu)$ are usefull for evaluation of $X(\lambda ,\mu )$ at oposite latitudes and multiple longitudes. Split in even and odd summation:

$$X^m(\mu) = \underbrace{\sum\limits_{n=\vert m\vert,+2}^N X_n^m P_n^m(\mu)}_... ...\underbrace{\sum\limits_{n=\vert m\vert+1,+2}^N X_n^m P_n^m(\mu)}_{X_o^m(\mu)}$$ (3.18)

$$X^m(-\mu) = X_e^m(\mu) -  X_o^m(\mu)$$ (3.19)

and use FFT to evaluate at multiple longitudes.

Triangular truncation

The ECMWF models uses a triangular truncation $M=N=T$, see figure 3.1.

Figure 3.1: Triangular truncation at $N=M=T$. Each dot $(m,n)$ denotes a coefficient $X_n^m$ of a spherical harmonic; the arrow denotes the order in which the coefficients are numbered. The coefficients for $m<0$ are often not in use, since for real valued $X(\lambda ,\mu )$ the coeffiecients satisfy $X_n^{-m}=\overline {X_n^{-m}}$.

Derivatives

Application of the nabla-operator to spherical harmonics (3.1) requires partial derivatives to $\lambda$ and $\mu$ (table 3.1):

$$\vec{\mathbf{\nabla}}X =  \frac{1}{a\cos\phi} (\ \frac{... ...l\lambda}  ,  (1-\mu^2)\frac{\partial X}{\partial\mu} )$$ (3.20)

Derivative in longitudinal direction:

$$\frac{\partial X}{\partial \lambda} = \sum\limits_{m=-T}^T \sum\limits_{n=\vert m\vert}^T (im X_n^m) P_n^m(\mu) e^{im\lambda}$$ (3.21)

Derivative in latitudinal direction:

$$(1-\mu^2)\frac{\partial X}{\partial\mu} = \sum\limits_{m=-T}^T \sum\limits_{n=\vert m\vert}^T X_n^m (1-\mu^2)\frac{\partial P_n^m(\mu)}{\partial\mu} e^{im\lambda}$$ (3.22)

$$= \sum\limits_{m=-T}^T \sum\limits_{n=\vert m\vert}^T X_n^m\left( (... ...n_n^m P_{n-1}^m(\mu) -  n \epsilon_{n+1}^m P_{n+1}^m(\mu) \right)e^{im\lambda}$$ (3.23)

$$= \sum\limits_{m=-T}^T\left( \sum\limits_{n=\vert m\vert-1}^{T-1} (n+2)\epsilon_{n+1}^m X_{n+1}^m P_n^m(\mu) \right.$$ (3.24)

$$\qquad\qquad\qquad\left.  -  \sum\limits_{n=\vert m\vert+1}^{T+1} (n-1)\epsilon_n^m X_{n-1}^m P_n^m(\mu) \right) e^{im\lambda}$$ (3.25)

$$= \sum\limits_{m=-T}^T \sum\limits_{n=\vert m\vert-1}^{T+1} \tilde{X}_n^m P_n^m(\mu) e^{im\lambda}$$ (3.26)

$$= \sum\limits_{m=-T}^T \sum\limits_{n=\vert m\vert}^{T} \tilde{X}_n^m P_n^m(\mu) e^{im\lambda}  +  \mathcal{O}(P_{T+1}^m)$$ (3.27)

where

$$\tilde{X}_n^m =  \left\{\begin{array}{lclcl} 0 & & &,& (T... ...\vert) \\ 0 & & &,& (m,\vert m\vert-1) \end{array}\right.$$ (3.28)

From spectral fields to grid

Quote from [Riddaway and Hortal, 2001], section 6.2:

It can be shown that, starting from the set of $2T+1$ complex Fourrier coefficients

$$X_{-T},\dots,X_{T} \qquad \in C$$

going to the set of $K$ grid points

$$x(\lambda_1),\dots,x(\lambda_K) \in R$$

and returning to $X_{-T},\dots,X_{T}$, we recover exactly the original values (the transforms are exact) as long as $K\ge 2T+1$ and the points are equally spaced in $\lambda$. This distribution of points with $K=2T+1$ is known as the linear grid.
On the other hand it can be shown also that the product of two functions can be computed without aliassing by the transform method of transforming both functions to grid-point space, multiplying together the functions at each grid-point and transforming back the product to Fourier space, as long as $K\ge 3T+1$. The distribution of points for which $K=3T+1$ is known as the quadratic grid.

Quote from [Riddaway and Hortal, 2001], section 6.6:

The integral with respect to the latitude can be performed from the Fourier coefficients by means of a Gaussian quadrature formula and it can be shown that this integral is exact if the latitudes at which the input data are given are taken at the points where

$$P^0_{N_G}(\mu) =  0$$

(these are called the Gaussian latitudes) with $N_G\ge(2T+1)/2$. Furthermore products of two functions can be computed alias-free if the number of Gaussian latitudes is $N_G\ge(3T+1)/2$. The Gaussian latitudes are not equally spaced as the points to compute the discrete Fourier transforms but they are nearly so and therefore this spacing is approximately the same as the longitudinal spacing.
The distribution of points allowing exact transforms is called the linear Gaussian grid and it has at least $2T+1$ longitude points equally spaced at each of at least $(2T+1)/2$ Gaussian latitude rows. Products of two functions can be computed alias-free if we use a quadratic Gaussian grid which is made of at least $3T+1$ equally spaced points in each of at least $(3T+1)/2$ Gaussian latitudes.
The same distribution of grid-points of a Gaussian grid can represent a linear or a quadratic Gaussian grid depend-ing on the spectral truncation used in conjunction with that grid. As an example, the quadratic grid corresponding to a spectral truncation of T213 coincides with the linear Gaussian grid corresponding to the spectral truncation T319.