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*To*: Frank Toussaint <address@hidden>*Subject*: Re: help with gaussian latitudes*From*: John Caron <address@hidden>*Date*: Wed, 02 Feb 2011 16:36:47 -0700

Hi Frank:

Thanks much for this background!

Thanks again for your time,

John

Hi John,

I came about this question a couple of years ago & needed some time to find someone who knows.

As you perhaps know, classical GCM (GlobClimMod) run on two grids: G1) spherical harmonics (SH) and G2) gaussian grids (GG).

G1) A transformation to spherical harmonics looks like a+b*sin[Lon]+c*sin**2[Lon]+ ..[lat] + ...). The nice thing is, that in the solutions of the dynamical equations you often have the second derivation - and this is very performant here: sin''(x) = -sin(x) --- you just switch the sign. However, in SH you always get global grids, as lat/lon need to run as: +-90/+-180. So the dynamical eqs are mostly solved on SH. Their variables (vorticity, divergence, ...) are often stored as SH files.

For other variables, SH are not a nice thing to use. So other equations are solved on a rectangular grid. But: as equations are coupled, you will need to run a million of conversions from one grid to the other and back.

G2) Using a GG as rectangular grid means, to take advantage of what good old Carl Friedrich Gauss (we write Gauß) found some centuries ago: there are ways of easy, nearly lossless conversion formulae between SH and a certain type of rectangular grids. The (small) price you pay is: strange latitude values. As far as I remember, they follow the zero-values of the Bessel function

http://en.wikipedia.org/wiki/Bessel_function

found by good old Friedrich Wilhelm Bessel.

Now the problem is that there are dfferent ways of conversion. Example: The old programme of www.ecmwf.int converted T63 to N48. At some day in the 80s, Hamburg (Max Planck Institute) got a copy of this programme. At some day in the 90s, ECMWF decided (after very [, very]**n lengthy in house discussions) to change their routines from, e.g., T63/N48 to T63/N32. So actually, e.g., "T128" means 2 different resolutions for most of the variables, depending on whether these 4 signs are at MPI or at ECMWF. This, of course, is only the case because most modellers use the SH resolution description for all the model instead of using something like T63/N48 or just N48 for all non-dynamical variables' resolutions. The difference between the two ways of conversion has to do with using the linear or quadratic coefficents in one of the transformation equations. This is why "TL" refers to "T with Linear conversion scheme" e.g. on pages

http://www.mad.zmaw.de/projects-at-md/era40/remarks/ (point 3.+4. of mine)

and

http://www.ecmwf.int/products/data/technical/gaussian/spatial_representations.html .

You find more interesting information from p 70 on in the appending document which I got from the ECMWF website.

In the grid transformation formulae (SH vs GG), the question which conversion to use is a question of degrees of freedom. Starting with SH, neither linear nor non-linear conversion has the same degree of freedom. One has a few less, the other a few more - this is why the conversion SH-GG-SH never is lossless. ==> Much room for mathmaticaldiscussions, which might be the better choice.

Hope this helps a little... and lets try to punish adequately any modeller using T<m> to describe the resolution of a GG...

Regards & a good weekend... frank

On 07.01.2011 14:51, John Caron wrote:On 1/7/2011 5:59 AM, Frank Toussaint wrote:Hi John, in old mails I just found your question. Please, let me know, wether this is still of interest. I think, I have some material. Regards... frank

On 11.12.2010 00:59, John Caron wrote:im looking at an grib1 file from Canadian Centre for Climate

Modelling, downloaded here:

http://cera-www.dkrz.de/WDCC/ui/Compact.jsp?acronym=CCCma_CGCM2_SRES_A2

the docs on that page says that "The atmospheric component AGCM2 is a spectral model with triangular truncation at wave no. 32 and 10 vertical levels." the actual grib file has 48 lats and the parameter N = "Gaussian Grid: N - number of latitude circles between a pole and the equator Mandatory if Gaussian Grid specified" equals 20.

My code thinks that with N = 20, the maximum lats should be 40. If anyone can shed any light or send me a reference I would be grateful. John

Hi Frank: Yes, this is still an open question. Can you shed any light? thanks, John

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