[CF-metadata] Proposal for spherical harmonics

From: Luis Kornblueh <kornblueh>
Date: Thu, 13 Mar 2003 14:29:30 +0100 (MET)

Hi,

we would like to include spherical harmonics as horizontal grid type
into the CF convention. This type of data is still very common with
NWP and some climate models. It is very efficient in terms of saving
space. I append our proposal which is in latex (sorry) and would like
to ask for comments.

Thanks a lot,
Luis

--
\documentclass[11pt,a4paper]{article}
\begin{document}
\title{Proposal for introducing the spectral representation in CF} 
\author{Luis Kornblueh\\Max-Planck-Institute for Meteorology}
\date{\today}
\maketitle
\section{Introduction}
An important class of representing atmospheric model results is given
by the spherical harmonic representation. This type of data
representation is non-local, but can be converted into grid-point
space by the Legendre Transformation.
An adequate way to describe spherical harmonics is given by the World
Meteorological Organization (WMO) in the Guide to WMO Binary Code Form
GRIB 2. We would like to include this description in the CF convention
without taking compression/encoding into account.
 
\section{Spectral representation}
Spherical harmonics should have a normalization so that they can be
used with associated Legendre polynomials of the first kind with a
normalization such that the integral over the associated Legendre
polynomials equals 1.
The associated Legendre polynomials of the first kind in general are
defined by:
\[
P_n^m(\mu) = \sqrt{(2n+1)\frac{(n-m)!}{(n+m)!}}
             \frac{1}{2^nn!}(1-\mu^2)^{\frac{m}{2}}
             \frac{d^{n+m}}{d\mu^{n+m}}(\mu^2-1)^n, \; \mbox{with} \; m \geq 0
\]
\[
P_n^{-m}(\mu) = P_n^m(\mu)
\]
A field $F(\lambda,\mu)$ in spherical harmonics is represented by:
\[
F(\lambda,\mu) = \sum\limits_{m=-m}^M\sum\limits_{n=|m|}^{N(m)}F_n^mP_n^m(\mu)
                 \mbox{e}^{im\lambda}
\]
where $\lambda$ is the longitude, $\mu$ the sine of the latitude, and
$F_n^{-m}$ the complex conjugate of $F_n^m$.  In the summations, $M$
is the maximum zonal wave number that is to be included, and $N(m)$
defines the maximum meridional total wave number $N(m)$, which, it
should be noted, is a function of $m$. Using two additional truncation
parameters $K$ and $J$ allows for the definition of different
truncation types.
\begin{center}
\begin{tabular}{lll}
Triangular  &  $M = J = K$ & $N(m) = J$   \\
Rhomboidal  &  $K = J+M$   & $N(m) = J+m$ \\
Trapezoidal &  $K = J, \; K > M$ & $N(m) = J$ \\
\end{tabular}
\end{center}
\section{Coefficient storage mode}
The complex numbers $F_n^m$ are stored for $m \geq 0$ as pairs of real
numbers Re($F_n^m$), Im($F_n^m$) ordered with $n$ increasing from $m$
to $N(m)$, first for $m = 0$ and then for $m = 1, 2, 3, \ldots, M$.
\section{Attributes required for horizontal representation}
For specifying the horizontal coordinate type use:
\begin{verbatim}
coordinates = "spectral"
\end{verbatim}
and required as global attributes:
\begin{verbatim}
dimension:
  :complex = 2
  :spc     = number of spectral coefficients
  :spcx    = associated number of longitudes
  :spcy    = associated number of latitudes
  :spherical_truncation_type = "triangular", "rhomboidal", or "trapezoidal"
  :spherical_truncation_n = N(m)
  :spherical_truncation_m = M
  :spherical_truncation_k = K
\end{verbatim}
It is assumed that only one type and resolution of spherical harmonics is
included in one file.
 
\end{document}
-- 
                             \\\\\\
                             (-0^0-)
--------------------------oOO--(_)--OOo-----------------------------
 Luis Kornblueh                           Tel. : +49-40-41173289
 Max-Planck-Institute for Meteorology     Fax. : +49-40-41173366
 Bundesstr. 55              
 D-20146 Hamburg                          Email: kornblueh at dkrz.de
 Federal Republic of Germany                                               
 Key fingerprint =  B3 69 B2 FC 81 65 F4 E7  44 45 EA 1A 82 BF D3 A6 

Received on Thu Mar 13 2003 - 06:29:30 GMT