The resolution of global spectral models

Abstract

Physics Depal1ment, University of Quebec at MontrealMontreal, Quebec, CanadaThe letter of Professor Pielke (1991) suggesting adefinition of horizontal resolution of gridpoint modelsprompts me to reciprocate with a short essay on acorresponding analysis for global spectral models.The horizontal structure of dependent variables inspectral models is represented by series expansion ofspherical harmonics Y;;', where 0 ::s Jml ::s n (e.g.,Kubota 1959); the transform method (Machenhauerand Rasmussen 1972) is used to calculate nonlinearterms on a Gaussian latitude-longitude grid in allmodern spectral models. Triangular truncation (O.$: n.$: N) of the series is often used, as it offers uniformresolution on a spherical domain. As this note will tryto show, there does not seem to be any straightfor-ward way to define an equivalent mesh size for spec-tral models; this makes the estimation of effectiveresolution even more difficult than in gridpoint models.A sometimes quoted estimate of spectral-modelresolution consists in the average spacing betweenGaussian latitudes of the transform grid; for triangulartruncation, this spacing is equal to that between longi-tudes at the equator: L1 = 21Ta/(3N + 1), with a theradius of the earth. Hence, L1 = 13.3/N in units ofthousands of kilometers; for a T31 model, L1 = 426 km.This estimate of resolution is overly optimistic becausethe dimensions of the transform grid are chosen toallow calculation of quadratic terms without aliasing ofthe resolved spectral fields, and hence the transformgrid is finer than required by the information content ofthe corresponding spectral series.A more realistic estimate of resolution is given bythe size of half a wavelength of the shortest resolvedzonal wave at the equator: L2 = 1Ta/N = 20/N in units ofthousands of kilometers. Hence, a T31 model wouldhave a resolution of L2 = 646 km, according to thismeasure. Triangular truncation provides an isotropicand uniform resolution on a sphere. The shortestresolved zonal wave (Iml = N) used to determine L2corresponds to a mode with the gravest meridionalstructure, because modes that are very short in onedirection must be elongated in the other direction:sectorial spherical harmonics (Iml = n ) have modalstructures shaped as orange segments and have theirlargest amplitude in the tropics, hardly an adequatemeasure of resolution for general circulation models.An alternative way to estimate the resolution ofspectral models is as follows. Consider that the area ofthe earth's surface is given by 47TB2; there are (N + 1)2real coefficients to a spherical harmonic series attriangular truncation with maximum index N, that is,N(N + 1 )/2 complex coefficients for the modes 1 :SIml :s N, plus (N + 1) real coefficients for the modeswith m = O. If an equal area on the surface of the earthis assigned to every piece of information contained inthe series, this gives a footprint of surface area equalto 41Ta2/(N + 1)2 for every real coefficient. Resolutionin a spectral model could be defined as the width L3 ofa flat, rectangular tile of the same surface area: L3 =(41T)1/2a/(N + 1) = 22.6/N in units of thousands ofkilometers. Hence, a T31 model would have a resolu-tion of L3 = 728 km, according to this measure.Yet another definition of resolution for spectralmodels would be to consider the representative spa-tial dimension of high-ordertesseral harmonics (0 < 1m I< n = N). The eigenvalue of the Laplacian operatorapplied on a spherical harmonics V':; is -K2 = -n(n +1 )/ a2. Equating the eigenvalue of the highest resolvedmode with the corresponding eigenvalue of Fouriermodes in Cartesian geometry for the purpose ofestimating resolution gives: K2 = N(N + 1 )/a2 = kx2 + ky2.Considering modes with unity aspect ratio, kx2 = k 2 =k2, that is, with checkerboard-like modal structOre,givesk2 = N(N + 1)/(2a2). An alternative measure ofresolution would be one-half of the correspondingwave-length: L4 = 1T/k = 21/21Ta/N = 28.3/N in units ofthousands of kilometers. So for a T31 model, L4 =899 km.It is noteworthy that the estimates of resolutiongiven by L2, L3' and L4 are all coarser than the simple-