### Introduction

This personal investigation represents a ‘see-for-myself’ look at the core concept of climate change known as radiative forcing (RF).

### Definition

The IPCC AR5 definition of RF is:

*“Radiative forcing is the change in the net, downward minus upward, radiative flux ( expressed in W/m^2 ) at the tropopause or top of the atmosphere due to a change in an external driver of climate change, such as, for example, a change in the concentration of carbon dioxide… The traditional radiative forcing is computed with all tropospheric properties held fixed at their unperturbed values, and after allowing for stratospheric temperatures, if perturbed, to readjust to radiative-dynamical equilibrium. …”*

The net radiative flux is of both long wave and shortwave radiation. The IPCC refers to RF as change from the 1750 atmosphere.

### Radiative Models

RF is determined using radiative transfer models. The Rapid Radiative Transfer Model ( RRTM ) is one used by many climate and weather models in the US and Europe. This look makes use of a somewhat older but easier to manage code, the Column Radiation Model ( CRM ) which was the original radiative code used for the NCAR Community Climate Model. The CRM was authored by Jeffrey Kiehl, Bruce Briegleb, and Charlie Zender.

### Myhre Paper

Much of the IPCC basis of RF derives from the 1998 paper by IPCC lead author Gunnar Myhre et. al. which refers to a 1997 paper by Freckleton et. al and a paper by Forster et. al. It is the Myhre paper that gives us the approximate 3.7 W/m^2 estimate of RF for a doubling of carbon dioxide and the associated simplified expression. Myhre cites the Freckleton use of the so called TREX atmospheric profiles as being necessary rather than just a single global average mean profile to calculate RF. The TREX atmospheric profiles are available in the pdf of the Freckleton paper. These soundings are idealized annual averages, one for the Northern Hemisphere, one for the Tropics, and one for the Southern Hemisphere.

### Mimicking Myhre

With a radiative transfer model and atmospheric profiles one may proceed to calculate RF. The CRM takes as input the time, location, pressure, temperature, water vapor, ozone, cloud fraction, cloud liquid water equivalent, snow cover, albedo, solar constant, and the composition of the most prevalent greenhouse gasses.

Radiative models use as input, the above parameters defined for vertical *levels*. The volume bounded by two levels defines a *layer*. The output of radiative model calculations includes upward and downward, long wave and shortwave RADIATIVE FLUX at each level expressed in Watts per meter squared. Also, for each layer, radiative models typically calculate the HEATING RATE, expressed in degrees Kelvin per day.

### Instantaneous Forcing

By executing the CRM model on the TREX soundings using a range of values for carbon dioxide from preindustrial to 1000ppm, one may examine the resulting radiative effect. The result of this calculation for the Tropical atmosphere is shown below in Figure 1:

### Adjusted Forcing

The cooling rate in the Instantaneous RF above is so rapid that a ‘fixed dynamical’ adjustment is applied, described by the Forster paper, by Ramanathan and Coakley and others. This adjustment can be achieved by iteratively computing radiance and applying a weighted portion of the resultant heating rate to the original temperature of the sounding until the iterated heating rate is smaller than a low limit. The result of this method for the Tropical atmosphere is seen in Figure 2 below:

### The Simplified Expression

Plotting the results of the adjusted forcing CRM runs over Figure 1 from the Myhre paper is exhibited below as Figure 3:

The so called ‘simplified expression’ of RF simply results from the logarithmic nature of radiative response due to the overlapping layers of a partially obscuring constituent. The IPCC identifies this as:

**RF = alpha * ln(C/C0)**

where **C** is a given CO2 concentration with respect to **C0**, the initial CO2 concentration ( pre-industrial or 279ppm for this look ).

Solving for a doubling results in alpha = RF / ln(2) or for these results, alpha = 5.27, close to the Myhre result of 5.35. The CRM uses cloud cover amount and my initial choice was 50% coverage for each of the layers. It turns out that choosing 25% or 75% would change the results. The more important aspect of my look was the process rather than a precise value, but the 50% cloud fraction is very close to what Myhre found.

**Comments**

So that’s where the simplified expression for CO2 forcing comes from. This exercise has been instructive to me and I hope it has been for you as well. Of course, RF represents a hypothetical – What if CO2 was suddenly doubled for a given atmosphere? And the atmosphere is represented by only three ‘average’ vertical profiles ( the TREX soundings ). In doingthe next post, I’ll show what I’ve found for additional ‘what if’ aspects of the question.

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