I. The simplest models are "dry dynamical cores". There is no moisture, and the thermodynamic equation is simply forced as a Newtownian relaxation to a prescribed equilibrium profile, which could be viewed as an idealized radiative-convective equilibrium.
II. The "Gray Radiation Aquaplanet Moist" GCM, or GRAM, introduces moisture as a prognostic variable, but explicitly excludes it from radiative transfer calculations. The introduction of moisture and a simplified radiation scheme requires a treatement of the surface and boundary layer. The model was developed by Dargan Frierson, Isaac Held, and Pablo Zurita-Gotor, and introduced in Frierson et al. (2006).
- The "gray" radiation considers a single longwave channel, governed by a time independent optical depth. The optical depth is set to approximate the impacts of the two key radiatively active gases: water vapor and carbon dioxide. Water vapor is concentrated near the surface and in the tropics by the temperature structure of the atmosphere, and this is reflected in a shallow drop off of the optical thickness.
- Incoming shortwave radiation is exclusively absorbed at the surface. In the standard configuration, the model runs in perpetual equinox, but this can be modified.
- The surface is treated a so-called "slab" ocean mixed layer, governed by single heat capacity that can be varied spatially, to crudely approximate land-sea contrast.
- The model includes a simplified boundary layer scheme, governing the exchange of momentum, heat, and moisture between the surface and atmosphere.
- The model can be run with (or without) a simplified Betts-Miller convection scheme. The greatest impacts are in the tropics, on both mean circulation and variability (waves activity), where true convection is decidely not on the grid scale.
III. The Model of an idealized Moist Atmosphere (MiMA), replaces the gray radiation scheme of GRAM with a full radiation scheme (RRTM), otherwise keeping the simplified hydological cycle of GRAM and the slab ocean lower boundary. A key simplification, however, is to assume a cloud free atmosphere. The model was developed by Martin Jucker, and introduced in Jucker and Gerber (2017), based on earlier work by Tim Merlis and coauthors, who used an alternative radiation scheme. See Martin's MiMA site on GitHub for more details and updates.
- As noted above, a key simplification of the model is to neglect microphysics: there are no clouds. Water interacts with the radiation scheme only in the vapor form. Overal, clouds cool the planet (it's actually a wash in the tropics, but midlatitude clouds are lower in the atmosphere, and so have a net cooling effect). To adjust for a realistic climatology, the surface albedo must be modified. In the default configuration we use a constant albedo, but one can have it vary with latitude to adjust for the cooling influence of clouds in the mid and high latitudes.
- Oceanic heat transport plays a key role in the tropics, where the atmospheric Hadley Cell is extremely inefficient. A specified oceanic heat transport within the "slab ocean", known as a Q-flux, was developed by Tim Merlis to produce a more realistic climatology.
- In the default configuration, there is now an annual cycle. One can fix radiation at a given date within the annual cycle, but this isn't exactly trivial: a planet left with solsticial forcing will settled down into a climatology that's quite more extreme than a January/July.
- The lower boundary conditions can be made more realistic. This paper led by Kevin DallaSanta (currently in review) illustrates what can be done.
IV. We can also GFDL's Atmospheric Model AM2.1, which was state of the art at the time of CMIP3. It includes a representations of clouds, aerosols, gravity waves, etc.. Contact me if you are interested in more realistic models -- we can also use NCAR's community climate models, and GFDL's HIRAM, which is built to run at higher resolution.