The Solar Series: I Background | II: The notch filter | III: The delay | IV: A new solar force? | V: Modeling the escaping heat. | **VI: The solar climate model (You are here)** | VII — Hindcasting | VIII — Predictions

**Open Science live — The story so far:** Dr David Evans is building the O-D notch-delay solar model. It’s a much simpler big-picture approach than Global Climate Coupled Models. They use an ambitious bottom-up system where the models add up every small aspect in every small cell of the Earth’s climate atmosphere and oceans and try to predict everything, but the trap is the errors — small errors in 10,000 calculations add up to big-mush. David’s approach is top-down. He looks at the whole system from the outside, and doesn’t try to understand or predict each individual part. It’s a way of starting at the start — to shed light on the big forces and processes that happen as energy arrives on Earth, gets reflected, or blended, and eventually changes the surface temperature. His model won’t tell us what happens to rainfall in Sudan in 2050, but it might do what current models don’t and that is *predict the global temperature.*

The important development here is to complete the path of the energy flow in the most brutally simple way from Sun –> Earth –> Space. We know the sun provides heat through TSI or Total Solar Irradiance. But this is almost constant — it produces heat for sure, but possibly not much of the variation in temperature on Earth that we are interested in. The discovery of the notch filter means some other force (yet to be specified) from the sun acts with a delay of probably 11 years. This delayed force turns out to cause a lot of the variation in temperature. But Earth is not going to immediately warm or cool with every change. Energy collects in all kinds of pools and buckets before it ends up warming the atmosphere. So the effects of both incoming paths — immediate solar and delayed solar — get combined and run through a “low pass” filter — which blends and smooths the bumps.

Having discovered the pattern in the way TSI is tranformed into temperature, David builds the model with the filters to produce the same “transfer function” as he found in empirical data. Hopefully the model will mimic the overall processes without needing to know the details of all the parts. In a sense all models have to do this at some level. No climate model tracks each molecule or follows each photon. Will it work? It does a good job of hindcasting (and we’ll talk about that soon), but the real test will take a few years. Enjoy the quest to figure it out.

By the way, one of my favourite graphs is below — Figure 4 — some curves are intrinsically beautiful. — Jo

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### Building a new solar climate model

Dr David Evans, 21 June 2014, David Evans’ Notch-Delay Solar Theory and Model Home

This is the last of the three posts in which we build the solar model. We assembled a notch filter, a delay filter, and a low pass filter in cascade in part III, in part IV we took a diversion to physically interpret the notch and the delay, and in part V we added the RATS multiplierto model the atmosphere on the yearly timescales of the TSI datasets.

In this post we assemble these four elements in their correct order, and add the immediate path for the TSI changes that obviously warm the Earth directly. This will complete the model. We finish by examining the step response of the model.

# The Order of the Filters

The notch-delay solar model so far is simply a computational path from TSI to (surface) temperature that contains a notch filter, a delay filter, a low pass filter, and the RATS multiplier (which is a trivial “filter” whose transfer function is a constant). There are no other filters we can discern from the empirical transfer function, or from elementary physical theory. So with no more to add, let’s put these four in order.

The transfer functions of these four filters, when multiplied together, form the empirical transfer function. The transfer function of two filters in cascade is the products of their two transfer functions, so these four filters must be in cascade (that is, the output of one is the input of the next). But multiplication is commutative, so the empirical transfer function does not indicate their *order*. For that we turn to physical reasoning.

The filter whose place is most obvious is the low pass filter. It models the Earth as a bucket of heat with unreflected TSI pouring in the top, and its output is the radiating temperature. We can now place the other filters around it.

In the flow of computation the RATS multiplier goes immediately after the low pass filter, because its input is the radiating temperature and its output is the surface temperature. We then have the computational path covered from the unreflected TSI all the way to the output of the entire model.

The notch and delay filters intrinsically go together and are inseparable, and it does not matter if they go notch-delay or delay-notch. The only place left for them to go is between the input to the entire model, namely the TSI, and the input to the low pass filter, which is the unreflected TSI.

Therefore the notch and delay filters are modulating the albedo of the Earth.

# The Immediate Path

The development to date only shows the *delayed* path from TSI to surface temperature. But obviously any changes in TSI also cause direct and immediate changes in the unreflected TSI, by changing the incoming heat from the Sun, so there is *also* an *immediate* path from TSI to the input of the low pass filter. This immediate path must therefore be in parallel with the notch-delay path from TSI to unreflected TSI.

# The Notch-Delay Solar Model

Putting it all together, here is the notch-delay solar model. If the recent global warming was associated almost entirely with solar radiation, and if it had no dependence on carbon dioxide, this is how it would work:

Note the parallel paths:

- The immediate path is for TSI, and has no effect on albedo. This is the direct warming effect of extra TSI.
- The delayed path is for force X, which is the same as TSI but delayed and notched. Force X affects the albedo.

The parameters for the model were found by fitting the model to the observed temperatures since 1610, when yearly TSI data became available, though focused mainly on the last 100 and 200 years. Composite TSI and composite temperature records were created out of the TSI and temperature records analyzed earlier. In forming the composites, the offset of each dataset was adjusted so that the average values for overlapping datasets are the same, datasets were faded in and out of a composite gradually rather than entering the average abruptly, and instrumental data was preferred over proxy data. The fitting process found the model parameters such that the model best reproduced the composite temperature from the composite TSI and best produced a transfer function like the empirical transfer function found earlier.

The most important parameter is the delay parameter, which was found to most likely be 11 years but definitely between 10 and 20 years. The break period of the low pass filter was found to most likely be 5 years, though the possible range is from 4 to 25 years because it might be hiding over to the low frequency side of the notch. (It is very unlikely to be more than about the five years that other researchers have found, but the fitting process held open the possibility.) The most likely set of parameters is called the “P25” set of parameters. The values in P25 were rounded off to form the “P0” set of parameters, which has been used to illustrate the transfer functions and step responses of the filters during this development.

Here is the transfer function of the entire model:

It reproduces the amplitude of the empirical transfer function (see Figure 5), in the grayed area.

Here is the step response of the model, in dark blue.

Note that the step response is causal — it is zero before the step stimulus is applied.

The step responses of each of the two paths in the model are also shown.

- The step response of the immediate path, in purple, is due to the direct warming effects of changes in TSI. It is small and has pretty much reached its final value after two years.
- The step response of the delayed path, in light blue, is due to force X. It constitutes the bulk of the overall response. It doesn’t reach its final value until about 15 years after the stimulus. Changes due to force X are not all exactly 11 years after the step change in TSI, but fade in gradually after just a couple of years, reach the crescendo of the “dagger” in year 11, then build to full strength after 15 years.

The final value of the delayed path response in the P0 parameter-set shown is 14 times larger than the final value of the immediate path response. The parameter fitting showed that this was the most likely value, and that the delayed path seems to always be between 10 and 20 times as powerful as the immediate path. Thus, the influence of changes in force X (or TSI via the the delayed path) on temperature is 10 to 20 times as powerful as the changes in TSI (or TSI via the immediate path).

The step response shown in Figure 4 is pristine and clean, with sharp edges, because it is a theoretical model, built of simple components that were inspired by, but do not incorporate, the messy empirical transfer function. If we could measure the step response of the system (TSI in, temperature out) then no doubt it would be lumpy, crinkly, and messy with no sharp edges, because in reality it is far more complicated than the model above. We aim to approximate a messy complicated reality with a simple model.

**All of the above and everything in the preceding posts are based on the solar assumption, that all the recent global warming is associated with TSI. Now that the parameter values have been estimated, we can dispense with the solar assumption.**

In the next post we will run some climate simulations, to see how well the model does at hindcasting.

Notch-delay solar project home page, including links to all the articles on this blog, with summaries.

**Jo adds that some people find Figure 4 looks unnaturally perfect (I did say it was my favorite). That’s true — it’s the model step response. The actual real one* *probably looks more complex. But there are no perfect “steps” either.*