Examples of Discretizations in Climate Models

In the ParCAL library we are creating the grid, or mesh, is a first-class citizen and we can calculate derived quantities taking the discretization in to account. Provided that the grid and discretization are adequately captured in the output files.

Sometimes the discretization used internally by the model is not what makes it out in to the history file.

Arakawa Grids

From Wikipedia: "The Arakawa grid system depicts different ways to represent and compute orthogonal physical quantities, notably velocity- and mass-related quantities, on rectangular grids used for Earth system models, notably for meteorology and oceanography. The five Arakawa grids (A-E) were first introduced in Arakawa and Lamb (1977)"

Figure from Rajpoot et al, J. Comp. Phys.

In the above, eta is a quantity such as temperature or moisture. u and v are velocities. Z-grid was introduced by Dave Randall.

The above figure is standard for showing the Arakawa grids but it can be confusing because it seems to suggest, for the A-grid, that everything is on a vertex. Here's another depiction:

Which better shows how these grids correspond to fields in the atmosphere.

Atmosphere Model CAM

The vertical and horizontal discretizations are treated separately.

Vertical Discretization

This is common to all the dynamical cores in CAM.

In the history file, the layer mid-points are indexed by "lev" and layer interfaces by "ilev". Lowest index value is the top of the atmosphere.

Horizontal Discretization - Spectral Element Method

From Dennis et al.: "To apply the spectral element method, we first decompose each eta-surface, assumed to be the surface of a sphere, using a quadrilateral finite-element mesh consisting of nonoverlapping elements. We assume the mesh is conforming (has no hanging nodes), and that each element can be mapped to the reference element with reference coordinates x and y."

Horizontal Discretization - Finite Volume

The FV dycore uses a D grid for all prognostic variables (see Lin-Rood, 1997). To avoid spurious two-grid-length gravity waves, the "time centered" velocities are calculated on a C-grid (denoted u* and v* in the figure below). The resulting grid is called the "CD-grid" in LR97.

In history files from CAM's FV dycore, most variables are provided in cell centers. the D-grid U and V components are also provided (as US and VS). The relationship between variables in the FV netcdf file and the grid for a 1-degree resolution FV dycore are illustrated below.

There are 2 ways of viewing the D-grid. The first (on the left) is as cell centers and edges. The second (on right) is 3 superimposed vertex-grids.