GEOS-Chem vertical grids

Reference

Sigma grid

The sigma grid is a purely terrain-following coordinate. The following met data products are defined on the sigma grid:

1. GMAO GEOS-3 (NOTE: Now obsolete)
2. GCAP/GISS meteroloogy
   • NOTE: GCAP met is really a hybrid grid, but it is defined as if it were a pure sigma grid (i.e. P_TOP=150 hPa, and negative sigma edges at higher levels.

The sigma grid is defined as follows:

           P(I,J,L)     - PTOP
   σ(L)= ------------------------
           Psurface(I,J) - PTOP

where:

   I,J,L         are the longitude, latitude, level indices of the grid box
   P(I,J,L)      is the pressure at a level edge or level center at grid box (I,J,L)
   Psurface(I,J)  is the surface pressure at grid box (I,J)
   PTOP          is the pressure at the top of the atmosphere

σ(L) always varies between 0 and 1, with 1 being the surface and 0 being the atmosphere top. Therefore you can think of σ(L) as the "fraction" of the atmosphere (measured from the top down) at which you are located.

The σ(L) therefore are independent of (I,J). The sigma values are typically fixed for a given met field type. Knowing the sigma value allows you to compute the center or edge pressures as follows:

   Pedge(I,J,L)   = σedge(L) * [ Psurface(I,J) - PTOP ] + PTOP

   Pcenter(I,J,L) = σcenter(L) * [ Psurface(I,J) - PTOP ] + PTOP

Any sigma grid can also be computed with the hybrid grid equation:

   Pedge(I,J,L)   = Ap(L) + [ Bp(L) * ( Psurface(I,J) - PTOP ) ]

   Pcenter(I,J,L) = [ Pedge(I,J,L) + Pedge(I,J,L+1) ] / 2 

where

   Psurface(I,J) = the "true" surface pressure at lon,lat (I,J)
   Ap(L)        = PTOP    = model top pressure
   Bp(L)        = σedge(L) = bottom sigma edge of level L

Hybrid grid definition

Earlier versions of the GMAO met data products used a pure-sigma grid definition. The problem with a sigma-grid is that you still see the signature of the mountains at the very top of the atmosphere. This was the case in GEOS-3 and this caused a lot of noise in the stratospheric winds, which led to poor STE.

The improvement on the sigma grid is the hybrid (or η) grid. This grid is defined with A and B coefficients, which are specified by the makers of the met data products (e.g. GMAO).

The following met data products use the hybrid grid formulation:

1. GMAO GEOS-4
2. GMAO GEOS-5
3. GMAO MERRA
4. GMAO MERRA-2
5. GMAO GEOS-FP

The pressure at the bottom edge of grid box (I,J,L) is defined as follows:

    Pedge(I,J,L) = Ap(L) + [ Bp(L) * Psurface(I,J) ]

    Pcenter(I,J,L) = [ Pedge(I,J,L) + Pedge(I,J,L) ] / 2

where

    I,J,L        are the lon, lat, level indices of the grid box
    Psurface(I,J) is  the "true" surface pressure at lon,lat (I,J)
    Ap(L)        has the same units as surface pressure [hPa]
    Bp(L)        is  a unitless constant given at level edges

The A_p(L) and B_p(L) for each met field type are listed in the GEOS-Chem source code file pressure_mod.f.

The hybrid grid formula in more depth

Rebecca Buchholz wrote:

I'm having trouble conceptualising exactly how GEOS-Chem deals with the vertical grid. I'm using the GEOS-5 reduced vertical resolution of 47 eta levels. I've seen the AMS definition of eta.

I can see the first fraction is the sigma definition (which follows terrain). Is the second fraction essentially a scaling by the pressure at altitude relative to pressure at sea surface? I'm not sure how it fits in with the GEOS-Chem definition, i.e. when rearranging the equation, what becomes A_p and B_p.

The manual and wiki pages indicate GEOS-5 is a hybrid pressure-sigma grid. Are the first 31 levels pure sigma levels in both the reduced vertical 47 and native 72 level GEOS-5 fields? Does this mean GEOS-5 doesn't use the above eta definition of the levels - only sigma definition? Does GEOS-Chem then change the sigma levels to eta levels?

Bob Yantosca replied:

Maybe we have used “eta” as a synonym for “hybrid” grid. Basically it refers to a grid that transitions smoothly from a sigma terrain-following coordinate near the surface to fixed pressure levels in the upper atmosphere.

Older met fields (e.g. GEOS-3) came on pure-sigma grids. However, the problem is that with a pure-sigma grid, you still see some signal of the mountains even at the top of the atmosphere. This makes the upper-atmosphere winds very noisy and it messes up the strat-trop exchange. (Or so GMAO tells us.)

The way I like to think of it is we specify the pressure at the bottom edge of grid box (I,J,L) with this formula:

   Pedge(I,J,L) = Ap(L) + [ Bp(L) * Psurface(I,J) ]

where Ap(L) and Bp(L) are specified by GMAO. Then you specify the pressures at the center of grid box (I,J,L) with this formula:

   Pcenter(I,J,L) = [ Pedge(I,J,L) + Pedge(I,J,L+1) ] / 2

i.e. it’s an average of the pressure at both edges.

Note that the Pedge(I,J,L) and Pcenter(I,J,L) vary w/ the topography up until about 170 hPa. That is where the first fixed-pressure levels occur. Skyward of that, then Pedge(I,J,L) and Pcenter(I,J,L) will be constant for all longitudes and latitudes (I,J).

Once you have computed Pedge(I,J,L) and Pcenter(I,J,L), you can construct an ETA coordinate such as:

   ETAedge(I,J,L)   = [ Pedge(I,J,L)   – Ptop ] / [ Psurface – Ptop ]

   ETAcenter(I,J,L) = [ Pcenter(I,J,L) – Ptop ] / [ Psurface – Ptop ]

but for the purposes of GEOS-Chem, we don’t really use the ETA values. We use the pressures computed from the Ap(L) and Bp(L) values.

The most important thing to note is that the thicknesses of the grid boxes near the surface vary with the surface pressure. When you come to a mountain, the levels all kind of bunch together and get small. When you are over flat land or ocean, the levels widen out.

Also note that a pure-sigma grid (like GEOS-3) can also be expressed using the same type of hybrid formulation:

  Pedge(I,J,L) = Ap(L) + [ Bp(L) * ( Psurface(I,J) – PTOP ) ]

where

  Ap(L) = PTOP (for all L)
  Bp(L) = Sigma_Edge(L) = the bottom sigma edge for level L

In GeosUtil/pressure_mod.f we use the same hybrid formula but we use the Ap(L) and Bp(L) that pertain to each grid (GEOS-3, GEOS-4, GEOS-5, GCAP, etc). That simplifies the coding.

--Bob Y. 15:58, 6 April 2011 (EDT)