In typical style I looked at this draft and told David that the second half of his post should be at the top (that’s where he discusses how his model solves so many problems). He replied that the equations were the most important part, and he wasn’t going to flip them around. So, for readers who don’t speak mathematica-lingua, all I can say, is don’t miss the second half below.

Also in typical style, David prefers this picture he’s just drawn in his diagramming software, to my cartoon in the intro to post 11:

In this post, David combines the two smaller models to make one basic climate model (that’s the sum-of-warmings and the OLR models). Unlike the mainstream conventional basic model that underlies the entire establishment culture and philosophy, the alternative model uses more empirical data (and from the real world too, not just the lab). It’s also less reliant on hypothetical partial derivatives. Plus, in the alternate model, different forcings can cause different responses. In the conventional model, the architecture assumes the climate responds to to all forcings the same way.

CO2 has a warming effect on the atmosphere, rather than just on the surface, and architecturally-speaking, this extra energy could be rerouted and escape through a different path (like water flowing through a different pipe). In the conventional basic model, remember, the only type of feedbacks allowed are responses to surface warming. The climate’s response is supposedly the same regardless of whether the Sun provides extra incoming energy or the atmosphere blocks a part of the outgoing flow. What was CO2 emissions shifts to become water vapor emissions. If the outgoing flow just redistributes from one wavelength to another , the conventional models are up a creek full of manure, so to speak.

The rerouting of outgoing energy is potentially a massive negative feedback, but in Certified Climate Model Speak we can’t even say that, since the term “negative feedback” is defined as negative feedbacks *to surface warming, not atmospheric warming*. In the blind language of standard models there is no term to describe this. No wonder they can’t get out of the rut they are stuck in.

To reduce problems with partial derivatives, David switches from using the Plank sensitivity to the Stefan Boltzmann sensitivity. The Planck variant relies on holding “everything else constant”, which allows the tropospheric temperatures to change (in unison) but keeps the stratospheric temperature constant. It’s different (slightly) from the Stefan Boltzmann sensitivity (which applies under all circumstances) principally because much of the CO2 and ozone emission layers are in the stratosphere.

We can do better than in 1896 — apparently Arrhenius used something akin to the conventional basic model, but with of course no climate data to work with.

— Jo

# 16. The Alternative Model

Dr David Evans, 29 October 2015, David Evans’ Basic Climate Models Home, Intro, Previous, Next, Nomenclature.

As discussed in post 11 on strategy, the alternative model is a combination of the sum-of-warmings model in post 13 and the OLR model in post 15. In this post we join them.

Some of the equations appear complicated, but they are simple — there are just a lot of factors.

### Joining the Sum-of-Warmings and OLR Models

We start with the sum-of-warmings model in Fig. 1 of post 13. However we only consider the solar and CO2 drivers, because more drivers introduce too many unknowns — so we omit the drivers marked “Other” in that figure. The analysis is between steady states, so the transitory effects of influences such as volcanoes may be ignored.

Adding the temperature perturbations due to ASR and CO2 in the sum-of-warmings model,

Applying energy balance (Eq. (1) of post 2),

Now join the models by using the OLR model, Eq. (14) of post 15, to replace Δ*R*:

A bit of re-arranging gives us the alternative model:

Equation (4) is called the ** joint-model equation (JME)**. It gives the surface warming in terms of the increases in the emission layer heights, the lapse rate, the cloud fraction, and the CO2 concentration. However it does so in terms of the CO2 sensitivity

*λ*

_{C}, which is still unknown at this stage.

### Estimating the CO2 Sensitivity

For a period between two steady states over which we have observations of the relevant climate variables, the JME estimates the * CO2 sensitivity* as

When *λ*_{C}, is known, the model is complete.

### Estimating the Fraction of Warming Caused by CO2, and the ECS

The ** fraction of global warming due to extra CO2** over the period between two steady states is estimated by the complete model to be

where Δ*T*_{S,C} is the surface warming due to CO2. The * equilibrium climate sensitivity (ECS)* is estimated as

### Diagram

There isn’t one, even though both constituent models have diagrams — Fig. 1 of post 13 and Fig. 2 of post 15 — because it is not possible to produce a single diagram such as those. In the sum-of-warmings model the CO2 influence *D*_{R,2X}Δ*L* *cannot* add to the ASR going into the solar response *Mλ*_{SB}. But in the OLR model, *D*_{R,2X}Δ*L must* add (negatively) to the OLR, which by energy balance is also the ASR.

### Comparisons with the Conventional Model

The alternative model here solves the problems of the conventional model described in post 5 and post 9:

- The sum-of-warmings approach allows for separate feedbacks for each individual driver. For example it allows for a CO2 sensitivity
*λ*_{C}that includes feedbacks that only apply to the influence of CO2, such as the rerouting feedback. The conventional model applies the same response to each driver, so it is structurally unable to accommodate influence-specific feedbacks such as the rerouting feedback. - In the sum-of-warmings model the drivers are not interchangeable. For example, it applies a different response to surface warming due to increased ASR than it does to decreased OLR from carbon dioxide molecules in the upper atmosphere due to CO2 enrichment. This undermines the notion or utility of forcing, because in the sum-of-warmings model the same forcing (radiation imbalance) from different drivers can cause very different surface warmings.

The alternative model ameliorates the problem withe conventional model described in post 4, because it has less reliance on partial derivatives and “holding everything else constant”:

- The sum-of-warmings architecture relies only on linearity to independently compute the warming due to each driver and to then add them, with no partial derivatives involved. In contrast, the radiation-balance architecture of the conventional model inherently depends on partial derivatives — the radiation that balances the forcings is computed by the Planck sensitivity, which relies on the Planck conditions (tropospheric temperatures change uniformly but everything else except OLR is held constant).
- The OLR model uses partial derivatives to account for the main factors that affect OLR. For example, it computes the change in OLR in the CO2 pipe by separately considering the effect of changing either the CO2 concentration, lapse rate, or surface warming while holding the other two factors constant. However the alternative model is less dependent on the value of the partial derivatives, and less vulnerable to errors in any one partial derivative, than the conventional model — because the conventional ECS is proportional to the Planck sensitivity
*λ*_{0}, a partial derivative. - The sum-of-warmings model employs the Stefan-Boltzmann sensitivity, which applies under all conditions. The heart of the conventional model is the Planck sensitivity, which only applies under the Planck conditions.

An advantage of the sum-of-warmings approach is that, with enough data gathered over enough time, it should be possible to statistically disentangle the warmings due to various drivers. In principle, the feedbacks and sensitivity to each driver could be verified by real-world data. In contrast, partial derivative estimates are intrinsically hypothetical and never empirically verifiable.

### Massive Negative Feedback Omitted from the Conventional Model?

Many who suspect that the ECS is substantially less than 3°C currently accept the conventional analysis for calculating the ECS (Fig. 2 of post 3). In the conventional basic climate model the only way the ECS can be dramatically lower than 3°C is if the total feedback *f* is much less positive than claimed in AR5, so that the feedbacks do not amplify the effect of a given forcing so much. Those critics have therefore searched for a massive missing negative feedback in response to surface warming. However the theoretical case for net-positive feedback to surface warming seems strong, based as it is largely on the properties of moist air, and after thirty years of searching it is getting unlikely that a massive negative feedback in response to surface warming has been overlooked.

The sum-of-warmings model (Fig. 1 of post 13) provides a possible answer for those critics: do not apply the solar sensitivity and feedbacks to the influence of CO2, but instead apply a separate CO2 sensitivity and feedbacks. Solar input heats the bottom of the atmosphere, changes occur in days, and it increases OLR, while CO2 enrichment affects outgoing energy from the higher atmosphere and changes apply over decades and it merely redistributes OLR between wavelengths and emitters (ignoring the minor albedo feedbacks to surface warming) — so might not they have different responses?

Although the rerouting feedback is not a “negative feedback” as that term is understood in the conventional model (because it is not a response to surface warming), if the CO2 sensitivity *λ*_{C} is much lower than the solar sensitivity *Mλ*_{SB} then it would have the same effect on ECS estimates as if a massively negative feedback had been omitted from the conventional model.