Re: help with gaussian latitudes

Hi Frank:

Thanks much for this background!

So you might be saying that N = 20 ("number of latitude circles between 
a pole and the equator") refers to the SH, which gets converted to a GG 
with N=24, which is reasonable.

If that seems like a possibility, I can code around the discrepancy. 
Otherwise I am of the opinion that the GRIB encoding is simply wrong in 
this case.

Thanks again for your time,

John

On 1/14/2011 10:55 AM, Frank Toussaint wrote:
> 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