# Improved Simple Climate Sensitivity Model

*by Richard J. Petschauer (February 24, 2011)*

**1. Summary of the Present Simple Model and Its Limitations**

The present simple climate model, used by most climate scientists to estimate climate sensitivity, is based on maintaining energy balance between the net solar incoming shortwave solar radiation and the outgoing longwave radiation from the planet. Increased CO_{2} reduces the outgoing radiation in the upper atmosphere where most of it originates, causing the temperature in the region of the radiation to increase until energy balance is maintained. While this seems adequate for CO_{2}, it ignores the required balance also of the atmosphere and the surface where feedbacks originate.

The present assumption is that the surface temperature increase will equal that at the emission level based on a constant lapse rate that defines the temperature difference between the surface and the emission level. A small reduction in the lapse rate with increased surface temperature has been estimated and is treated later as negative feedback and combined with other feedbacks. The IPCC estimates this as = – 0.84 Wm^{-2 }/ C (vs. water vapor at +1.8 Wm^{-2} / C for example).

The temperature sensitivity factor, T_{s}, is the change in temperature associated with a change in radiation based on the radiation equation W = seT^{4 }in units of Wm^{-2},with T in K, s = 5.67e-8 and e, the emissivity, is treated 1 for saturated gases and clouds. Based on the this it can be shown,

(1.1) T_{s} = 1 / (4W ^{0.75} s ^{0.25})

Which can be approximated as,

(1.2) T_{s} = 16.20 / W ^{0.75 }

With energy balance at the top of the atmosphere (TOA) now estimated at 235 Wm^{-2},

(1.3) T_{s} (TOA) = 0.2699 C / Wm^{-2}

This compares to the value at the surface of 15 C and 390 Wm^{-2}

(1.4) T_{s} (S) = 0.1846 C / Wm^{-2 }

With an unbalance at the TOA from doubling CO_{2} estimated at 3.71 Wm^{-2},

(1.5) DT (2x CO_{2}) = 0.2699 x 3.71 = 1.00 C (before feedbacks)

**2. Basis and Advantages of Improved Simple Model**

The proposed new model adds the requirements of energy balance at the surface and the atmosphere as separate objects. It can be shown that if the constraint that the atmosphere is balanced regarding energy in and out, then the surface will also be balanced. The advantage of this new model is that other factors besides CO_{2 }changes can be included with the most important estimates being the role of evaporation changes with surface temperature. This model is based on well established estimates of the present mean global energy balance by Kiehl and Trenberth and others.

Figure 1 shows results from a 1997 paper and Figure 2 is an update from 2009. The values in Figure 1 are in balance, while those in Figure 2 are not. Because of this for purposes of explaining

the new model, we will use the values from Figure 1. The 2009 values require a solution by the model to represent the balanced state as a new baseline. Later we show using either set of data gives nearly identical results.

**3. Improved Simple Model Equations**

Figure 3 is a schematic representation of the values from Figure 1 with the cloud and surface albedo left out with just the net solar, S, shown. From Figure 1, the present balanced conditions before any perturbation changes are (all in Wm^{-2}),

- S = 342 – 77 – 30 = 235
- A = 67
- H = 24
- E = 78
- G = 390
- W = 40
- a = (390 – W) / 390
- k = 195 / (195 + 324)

where W is the amount through the atmospheric window in cloud free areas.

Balance at the top of the atmosphere (TOA) as in the present model is required. The primary new feature is that we break the outgoing longwave radiation of 235 Wm^{-2} into three components and their relationship to heat flux inputs to the atmosphere, some of which are not related to longwave radiation. For balance of heat flux in and out at the TOA we get,

(3.1) S = k (A + H + E) + kGa + G (1 – a)

Solving for G,

(3.2) G = (S – k (A + H + E)) / (1 – a + ak)

We calculate the surface radiation absorption factor, ** a**, from the total radiation from the surface minus the amount through the atmospheric window,

**, with a value of 40 in Figure 3.**

*W*(3.3) a = (390 – 40) / 390 = 0.8974

The outgoing fraction,* k*, is the outgoing flux from the atmosphere divided by the total input to the atmosphere,

(3.4) k = U / (A + H + E + G – W)

. = 195 / (67 + 24 + 78 + 390 – 40)

. = 195 / 519 = 0.3757

where ** U** = the upgoing radiation from the atmosphere including clouds of 195. Note also

(3.5) U = S – W at balance

The downwelling radiation (“back radiation”), ** D**, = the total absorbed by the atmosphere minus the upgoing, or 519 – 195 = 324. Or it can be calculated directly for balance at the surface,

(3.6) D = H + E + G – (S – A) = 24 + 78 + 390 – (235 – 67) = 324

So the downwelling can be thought as a dependent variable, not the cause of greenhouse warming. However, it is important to note that any changes in the independent variables of H, E, A, or S will cause D to change as will changes in G which varies as a balancing reaction to the independent variables and to changes in absorption, a, all based on the above equations. In summary, both G and D are forced to change in response to perturbations of the primary parameters.

If we know both U and D from estimates such as in Figure 1, then a simple expression for k is,

(3.7) k = U / (U + D) = 195/ (195 + 324) = 0.3757

Note that this model assumes the outgoing fraction, k, does not change significantly. Discussion on this appears in Section 6.

**4. Basic Results Using Improved Model Equations**

**4.1 Longwave Forcing at the Top of the Atmosphere (TOA) **

For this we estimate the climate temperature sensitivity factor that relates changes in longwave unbalance at the TOA, T_{s} (TOA) in C / Wm^{-2}. To do this we force a 1 Wm^{-2} reduction in the outgoing radiation and determine a new surface radiation, G, a new value of T_{s} at the surface and a resulting temperature change. We can do this in several ways, one of which is to reduce the atmospheric window factor from 40/390 to 39/390 which changes the atmospheric absorption, a, to (390 – 39) / 390 or 0.9000.

Using equation 3.2, we get a new value of G at 391.4248, a change of 1.428 from G_{0} of 390, the original value. This represents a surface temperature change of

Ts(TOA) = DT = (391.1428 ^{.25} – 390 ^{.25}) / s ^{.25} = 0.2627 C / Wm^{-2 } (4.1)

This compares closely to the estimate from the present method of 0.2699 from equation 1.3. If one solves for the new value of U, the outgoing radiation, it increased by 0.8575, less than the 1 Wm^{-2} forcing, because W has increased slightly in response to the increased surface radiation, such that the increase in U + W = 1, the value initially lost through the window.

If we use TOA forcing of 4 Wm^{-2 }and divide the result by 4, T_{s} (TOA) increases slightly to 0.2645 C / Wm^{-2}, indicating a small nonlinearity.

**4.2 Shortwave Solar Forcing at the Top of the Atmosphere (TOA) **

Using (3.2) with S increased from 235 to 236 and A by 236/235 to maintain the same relative fraction, gives a new value of G of 392.0304, corresponding to a surface increase of 0.3741 C, substantially larger than 0.2627 C from longwave forcing.

**4.3 Forcing at the Bottom of the Atmosphere (BOA) **

This is a method to estimate the effect of surface evaporation increasing with surface temperature since the present simple model assumes it does not change or that the lapse rate somehow compensates for the increased latent heat moved to the atmosphere when clouds form. The increased evaporation will result in negative feedback. Sensitivity at the BOA can also be used to estimate changes (feedback) from increased water vapor at lower altitudes.

We do this by changing the value of E by 1 Wm^{-2 }and calculate the surface temperature change. Using equation 3.2, we get a new value of G at 389.456, corresponding to a surface temperature drop of 0.1579 C. (We estimate the original temperature based on 390 Wm^{-2 }at 14.9853 C). So,

(4.2) T_{s} (BOA) = -0.1579 C / Wm^{-2}

If we add 1 Wm^{-2} forcing at the TOA and then 1 Wm^{-2} of evaporation increase, the temperature decreases 0.1580 C, nearly the same amount.

**4.4 Estimating Evaporation Negative Feedback from T _{s} (BOA)**

Complex computer climate models estimate evaporation rates increase about 2 – 3 % / C, while basic physics indicate about 6% / C with the typical assumption of increased water vapor so as to maintain constant RH. Data over oceans by Wentz et al indicates a rate close to 6% / C. This new improved simple climate model can be used for any rate change.

There are two methods. In this section we will explain the simpler one and use the other method as verification. The previous section estimated a temperature sensitivity factor at the bottom of the atmosphere of 0.1579 C / Wm^{-2}. For *r*, the evaporation in % / C, the *feedback value, F*, based on the present value of evaporation at 78 Wm^{-2} is r / 100 x 78 x -0.1579.

(4.3) For r = 2.5% and 6%, this results in feedback values of -0.308 and -0.739 C / C.

This compares to IPCC (that has no feedback for evaporation changes), estimates for water vapor positive feedback of 1.8 Wm^{-2} / C at the TOA where they use a Ts(TOA) of about 0.300 C / Wm^{-2} resulting in a feedback value of + 0.504 C / C. So it appears that it is likely that negative feedback from surface evaporation can readily offset all or a major portion of the estimated positive water vapor feedback.

**4.5 Estimating Evaporation Negative Feedback from the Final Balanced State**

Equation 3.2 can be modified to allow E, the evaporation rate to vary with the change of surface temperature implied from the change in G, the surface radiation. Let

(4.4) X = S – k (A + H) – k (E – G_{0}ET_{s}r/100),

and

(4.5) Y = 1 – a (1 – k) + kET_{s}r/100,

then

(4.6) G = X / Y

the new surface radiation. The final value of evaporation,

(4.7) E_{f} = E + T_{s}E (G – G_{0}) r/100

Where r = the rate of change of surface evaporation in % / C, E the initial evaporation, G_{0} the initial surface radiation, and Ts the surface temperature sensitivity factor at G_{0} per equation 1.1 or 1.2.

Earlier we estimated the case of 1 Wm^{-2} longwave forcing at the TOA where T_{s} for constant evaporation was 0.2627 C. Using equation 4.6 with r = 6%, the value drops 0.1510 C. This indicates a ** feedback multiplier, m** of 0.1510 / 0.2627 or 0.5748. F is the sum of the feedbacks (in this case there is only one). Then, the feedback multiplier,

(4.8) m = 1 / (1 – F)

Solving for F gives,

(4.9) F = (m – 1) / m = (0.5748 – 1) / 0.5748 = -0.740 C / C

which compares closely to -0.739 from section 4.4.

**5. Comparing Results Using Updated Energy Balance Estimates from 2009**

Since the data from Trenberth et al shown in Figure 2 is not in balance, the balanced value of G, the surface radiation, must first be determined using equation 3.2. This results in an increase from 396 to 398.259 Wm^{-2} resulting in a surface temperature of about 16.5 C. The results below compare the 1997 / 2009 values. The units are all in C / Wm^{-2} except the feedback values which are C / C.

- Ts (Surface): 0.1846 / 0.1815
- Ts (TOA to Surface) at 4 Wm
^{-2}longwave forcing: 0.2645 / 0.2646 - Solar forcing from net 1 Wm
^{-2}solar increase: 0.3741 / 0.3647 - Ts (BOA) at 1 Wm
^{-2}increase in evaporation: -0.1579 / -0.1555 - Feedback value with evaporation increase of 2.5 % / C: -0.308 / -0.311
- Feedback value with evaporation increase of 6 % / C: -0.739 / -0.746

**6. Is the Assumption of Fixed Up Going Atmosphere Radiation Fraction Valid? **

Central to our method used here is a fixed fraction of flux leaving the atmosphere going out to space relative to total absorbed by the atmosphere, unless the perturbation is a change in this ratio. We think of changes in the fraction caused by the balancing process as additional feedback, similar to the lapse rate feedback presently used. The model itself can give this change if we know how the up going fraction changes. At this time we have no way of doing this except in a qualitative way that judges if the fraction increases or decreases. If it increases, the surface temperature change will decrease, the opposite will increase it.

For the case of a forcing of increased downward flux at the TOA, we think the outgoing fraction will increase causing added cooling, so our estimate gives an upper bound of the final temperature rise. For the case of evaporation cooling, we think that the heat transferred when the water vapor condenses will be at a higher altitude compared to where the typical longwave radiation from the surface is absorbed that we estimate from other work at about 50% by 1 km. This will increase the up going ratio and increase the magnitude of the negative feedback, so we think our estimate is conservative. Also at altitudes of cumulus cloud tops, water vapor is greatly reduced and the most CO_{2} captures is about 20 to 22% of the radiation leaving upward from the cloud top.

This means that the atmospheric window with clear skies above could be up to 80% of the total radiation, compared to only about 25% up from the surface. And Kiehl and Trenberth (1997) estimate above about 6 km, cloud cover is only about 20%. Furthermore, increased high altitude clouds will increase solar albedo, adding to the negative feedback.

**7. Comparison to IPCC Estimates**

Including the evaporation negative feedback into the IPPC estimates for 2x CO_{2} surface warming reduces the estimated temperature increases substantially. The IPCC 2007 numbers including all feedbacks are: a central value of 3.2C with a range from 2 C to 4.5 C. Details of how this estimate was made were not disclosed, but literature research by Monckton (2008) referencing Soden and Held cite these feedback factors:

- Water vapor = +1.8;
- Lapse rate = -0.84;
- Clouds = +0.69;
- surface albedo = 0.26;
- CO
_{2}= 0.25.

These sum to 2.16 Wm^{-2} / C. The feedback value is obtained by multiplying this sum by the temperature sensitivity factors at the TOA, a value that IPCC did not disclose. Back calculating using the feedback factors and a temperature rise of 3.2 C, indicates a value of T_{s} of 0.301 C / Wm^{-2}. This gives a central estimate of the feedback value at 2.16 x 0.301 or 0.6502, with a range from 0.4416 to 0.7518.

However, we think the value of T_{s} of 0.301 is too high, and a more correct value is 0.2645 as indicated in section 4.1 This lower value of T_{s} reduces both the pre-feedback temperature increase and the feedback value.

Table 1 compares the estimates using the IPCC feedbacks and their value of T_{s}, and also the reduced T_{s} value we estimate. Table 1 also combines these IPCC corrected feedback values with our evaporation negative feedback values of -0.308 and -0.739 for 2.5% and 6% / C changes in evaporation rates, with plus/minus 10% tolerance added to these.

There has been a question by Roy Spencer and others about the value for cloud feedback, even whether it is positive as IPCC estimates. Table 1 also shows the results with the evaporation feedbacks included and no cloud feedback, and finally with negative cloud feedback of the same magnitude as that of the positive value. Before feedback, IPCC = 1.12 C, this estimate = 0.98 C.

Note with negative feedback the relative temperature variations are greatly reduced.

Max / min drops from 2.25:1 down to 1.29:1.

**8. Most of the Evaporation Negative Feedback will be delayed by Slow Ocean Warming **

Most of the global evaporation originates over the oceans, and much of the precipitation over land is reevaporated. Any global warming will first be seen first over land, but the recycling fraction will increase with temperature from both the surface and vegetation, so some negative feedback will exist.

The major negative feedback will be delayed until the oceans warm, reducing the rate of temperature increase compared to that before this time. From this it follows that rather than future large heat being stored in the oceans, causing a potential delayed “tipping point”, stabilizing negative feedback should be expected.

**References**

Kiehl, J. T., and K. E. Trenberth (1997): Earth’s Annual Global Mean Energy Budget. *Bull.* *Amer. Meteorol. Soc.,* **78**: 197-208

Kiehl, J.T. (1992) *Atmospheric general circulation modeling*, in *Climate System Modeling,* ed.

Mitchell, J. F. B., C. A. Wilson and W. M. Cunnington: (1987) On CO_{2} climate sensitivity and model dependence of results. *Q. J. R. Meteorol*. Soc. **113**, pp. 293-322

Monckton, Christopher (2008), Climate Sensitivity Reconsidered, *APS Physics Newsletters *

Soden, B.J., and Held, I.M. (2006) An assessment of climate feedbacks in coupled ocean-atmosphere models. J. Clim.**19:** 3354–3360.

Trenberth, K. E., Fasullo, J. T. and J. T. Kiehl (2009): Earth’s Global Energy Budget, *Am. Meteorol. Society, *March 2009

Wentz, F. J., L. Ricciardulli, K. Hilburn and C. Mears: (2007) How much more rain will global warming bring? *Science*, **Vol 317**, 13 July 2007, pp. 233-235

E = MC^2

Too simple?