/sci/ - Science & Math

>>8711279
Not sure what you meant by "dimensionally consistent."

I take that's the empirical equation for climate sensitivity used in the GISS GCM model. It is just an empirical fit based on simpler 1D model, described in detail here:
http://shadow.eas.gatech.edu/~kcobb/warming_papers/Hansen_etal_1984.pdf

The basic idea is this. Energy in must equal energy out, 2nd law of thermodynamics. In 1984 paper they did a 1D model assuming various CO2 concentration (and other greenhouse gases too), what would happen if you double or triple or quadruple CO2, what change in temperature of earth needed so that the energy equation balances. You received a finite amount of energy from the sun, given at solar constant. This energy is absorbed by earth 100% since earth is essentially a blackbody. Then the energy is re-emitted by the Earth following stephan boltzmann equation energy out = sigma*T^4. T here represent temperature of earth's surface, sigma is stephan boltzman constant. Energy out is simply a single function of Earth's temperature. Now here's the slightly tricky part. A FRACTION of this energy out is reabsorbed in the atmosphere and reflected back to earth. If you increase greenhouse gas concentration in the atmosphere, you're gonna get more IR reflected back to the surface, because your reflectance got "thicker". You basically integrate from bottom to top of atmosphere, assuming every layer of atmosphere absorbs and re-emit infrared radiation continuously based on prescribed concentration gradient (based on partial pressure of said molecules) with infinitesimal dz. In the very top of the atmosphere the energy must balance, energy from the sun = energy leaving the earth. Then you solve for T earth, that's your equilibrium T for a given greenhouse gas concentration.

They then take an empirical relationship for a given concentration range from this 1D model and use it in the 3D GISS model as climate sensitivity parameter