# Looking for the mathematics so I can run climate change simulations

Sorry for the stupid question, but I have been unable to answer it even after wasting time googleing for a while... I know there are several climate-changes scenarios, and I know the several climatic databases offering surfaces (rasters) with those scenarios. However, I want to try some calculations on my own, for which I am looking for the mathematics behind these models (the different scenarios) or some code (preferably in R). Any suggestion?

• If you are interested in a basic understanding and not in exact results, I suggest to look at basic concepts of climate physics like equilibrium climate sensitivity, the radiative equilibrium and radiative forcing. You can easily estimate for example earths temperature without greenhouse effect. There are probably many example calculations on this site as well. See e.g. my answer here (not directly related to your question though): earthscience.stackexchange.com/questions/20937/… + comments below. Nov 5, 2021 at 13:18
• I could've sworn there was a post on some Stack network (perhaps not this one) about getting the nobel prize winner's early model running on your machine... but gosh if I can't find it. Simple climate models that predict climate change may offer little bits of use. Nov 6, 2021 at 3:47
• These also look interesting: Simple Climate Model Lab and this Reddit thread on climate modeling in R Nov 6, 2021 at 3:51
• @JeopardyTempest Suki Manabe ftw: I thought it previously had an answer, but apparently not. Nov 6, 2021 at 10:44
• If you're seriously interested, try Raymond PierreHumbert's "Principles of Planetary Climate". I don't think you're likely to find serious models written in anything but C/C++ or Fortran. Nov 7, 2021 at 3:35

# Full scale climate models

The atmospheric climate simulations that are presented in the IPCC reports are performed by quite large and complex models. Running these models needs a "supercomputer" (high performance cluster). Additionally, certain "forcing data" and "boundary conditions" are needed.

Boundary conditions are the top of the atmosphere and the Earth's surface/oceans. Partly, atmospheric climate model simulations are directly coupled to ocean model simulations. This makes the models even larger and harder to setup/run. "Forcing data" means thinks like CO2 emissions, emissions of particulate matter, solar radiation ... .

Having said that: it is quite unrealistic to setup und run a "real" climate model just for fun.

# Further reading on recent climate modeling activities

Simulation results of the couple model intercomparison project (CMIP) form the basis for the IPCC reports. The CMIP Phase 6 (CMIP6) simulations form the basis of the most recent IPCC reports. There is an overview paper of CMIP6 which might indicate how large the overhead for running "real" climate models in a comparable way is (Eyring et al., 2016).

One of the models that were used for the CMIP5 and CMIP6 simuations was the MPI-ESM model. An overview of this model is given in Giorgette et al. (2013). Detailed descriptions are available in this special issue.

There are several more models used in CMIP5 and CMIP6. The MPI-ESM is just an example.

# Climate models made for training

There are some simple climate models that are made for students and which are based on a few governing equations. Seems models show the general features of climate models but can be run on a normal end-user computer. One of these models is the "Monash simple climate model". An instance of that model which can be run via a web-GUI is available here. There is a documentation available in which you will find links to a publication describing the model and to a repository to download the source code.

There is another climate model made for training which is called Planet Simulator. Download details and documentation are offered on that web page.

• I find it quite confounding that the physical reason for the greenhouse effect, i.e. the increase in infrared optical depth $\rm \tau_{IR}$, and simple solutions for the temperature $\rm T(\tau_{IR})$ such as discussed in ui.adsabs.harvard.edu/abs/2010A%26A...520A..27G/abstract are never to be found in the terrestrial climate literature. The only thing to be found are two-stream like 1-layer models with the emissivity and absorptivity being given as magical numbers. Is there a reason for this in the geo community? Did stellar and terrestrial atmospheric physics diverge sometime? Nov 8, 2021 at 18:01
• @AtmosphericPrisonEscape I have no idea. Nov 8, 2021 at 23:32

You might try a latitudinally-resolved climate model. These were invented in 1969 to see how global warming or cooling moved the boundaries of the polar caps. One result is that a slight decrease in the Solar constant glaciates the Earth; that's because the models don't include the carbonate-silicate cycle which stabilizes Earth's temperature in the long run. But for examining small changes it might be okay.

The basic equation is one of power balance (usually incorrectly called "energy balance"):

Qi (1 - Ai) = A + B Ti + C(Ti - T)

Here Qi is the insolation, averaged over the year, in a given latitude band, band i--you need a lot of spherical astronomy and trig to figure this out, but some articles (e.g. Warren and Schneider 1979 or Chylek and Coakley 1975) have precalculated tables you can use. Ai is the albedo in that band--you can set this to flip from a high value (bright, icy) to a low value (dark, uncovered) when temperature goes over a critical value, like 273 K. The whole left-hand side of the equation, then, is the solar power absorbed in that climate band.

A + B Ti is the infrared power emitted by that band to space. A and B are constants (A is not the albedo Ai), and Ti is the band's temperature--normally you start out with an isothermal Earth and then iterate until the simulation stabilizes. Values for A and B vary somewhat in the literature, but in a recent paper I used A = 214.5 W m^-2 and B = 1.57 W m^-2 K^-1.

The last term is the energy moved from one band to another, in a very simple (read "crude") "diffusion" model. C is a constant, Ti is and temperature, and T is the average temperature of the whole globe (I'll tell you how to calculate that in a minute). Again, values for C vary. Recently I used C = 3.74 W m^-2 K^-1.

You can find the mean global albedo or temperature or what-have-you by multiplying the zonal values by the zonal area fraction and adding up all the figures. You find the fraction of a hemisphere's area by subtracting the sine of the more equatorial boundary from the sine of the more polar boundary. For instance, the area of a hemisphere between 30 and 40 degrees is sin(40) - sin(30) = 0.1428.

P. Chýlek, J.A. Coakley, Jr. J. Atmos. Sci. 32, 675-679 (1975).

S.G. Warren, S.H. Schneider. J. Atmos. Sci. 36, 1377-1391 (1979).