Solar Cycle 24 peaked? The experimentum crucis begins.

The WSO Polar field strengths – early indicators of solar maximums and minimums – have dived towards zero recently, indicating that its all down from here for solar cycle 24.

Polar field reversals can occur within a year of sunspot maximum, but cycle 24 has been so insipid, it would not be surprising if the maximum sunspot number fails to reach the NOAA predicted peak of 90 spots per month, and get no higher than the current 60 spots per month.

The peak in solar intensity was predicted for early 2013, so this would be early, and may be another indication that we are in for a long period of subdued solar cycles.

A prolonged decline in solar output will provide the first crucial experiment to distinguish the accumulation theory of solar driven temperature change, and the AGW theory of CO2 driven temperature change. The accumulation theory predicts global temperature will decline as solar activity falls below its long-term average of around 50 sunspots per month. The AGW theory predicts that temperature will continue to increase as CO2 increases, with little effect from the solar cycle.

An experimentum crucis is considered necessary for a particular hypothesis or theory to be considered an established part of the body of scientific knowledge. A given theory, such as AGW, while in accordance with known data but has not yet produced a critical experiment is typically considered unworthy of full scientific confidence.

Prior to this moment, BOTH solar intensity was generally above its long term average, AND greenhouse gases were increasing. BOTH of these factors could explain generally rising global temperature in the last 50 years. However, now that one factor, solar intensity, is starting to decline and the other, CO2, continues to increase, their effects are in opposition, and the causative factor will become decisive.

For more information see WUWT’s Solar Reference page.

Phase Shift in Spencer’s Data

It was shown here that the phase shift between total solar irradiance and global temperature is exactly one quarter of the solar cycle, 90 degrees, or 2.75 years. This is a prediction of the accumulation theory described here and here that shows how solar variation can account for paleo and recent temperature change.

Phase shifts in the short-wave (SW) side of the climate system are erroneously attributed to ‘thermal inertia’ of the ocean and earth mass, and called ‘lags’, or regarded as non-existent. If thermal inertia was responsible, then a larger mass would show a larger lag. In fact, an exactly 90 degrees shift emerges directly from the basic energy balance model, C.dT/dt=F, as I will show later.

A 90 degree shift is also present on the long-wave (LW) at the annual time-scale using Spencer’s dataset. This cannot be a coincidence, and gives an important insight into the dynamics of the climate system.

First off is understanding how shifts arise.

The figure above shows an impulse (black) based on a cosine function with a 2*pi period, with its scaled derivative (green) and integral (red). Time is on the x axis.

The impulse in black represents any sudden change in forcing in the atmosphere that ’causes’ the derivative and integral responses (as they are derived directly from the impulse).

Note two things: (1) the peak of the derivative leads the peak of the impulse, and the peak of the integral lags the impulse. (2) The lead and lag are exactly one quarter of the period (2*pi/4 or 1.57 radians) of the cosine impulse. Note (3) the integral ‘amplifies’ the impulse, the mechanism responsible for high solar sensitivity in the accumulative theory.

Cross-correlation (ccf in R) of two variables gives precise information about the phase shifts, their size and significance. Above is the cross-correlation of the derivative and integral with the impulse above, with significance as blue lines. You can read off the phase shift from the first peak location.

The data from Spencer consists of satellite measurements of the short-wave and long-wave intensities at the top of the atmosphere, both for clear sky and cloudy skies. Below is the cross-correlation of each of these variables against his global temperature HadCRUT3 column.

The peaks of correlation show a three month phase shift on the LW and SW_clr components. The LW peaks are positive and the SW peaks are negative due to the orientation of flux in the dataset.

The LW peaks (LW_tot and LW_cls) are affected by the sharp peak at zero lag, probably due to fast radiant effects (magenta line SW_clr), shown in the similar graphic of these data by P.Solar here, mentioned in this thread at CA.

The LW and SW_clr components lead the global surface temperature. There are three possible explanations:

1. Changes in cloud cover actually do drive changes in global temperature due to gamma-ray flux (GRF) or other effects, or

2. The changes in cloud cover are caused by changes in global temperature, with the derivative mechanism described above.

3. Both 1 and 2.

Spencer argues that it is impossible to distinguish between 1 and 2. Both Spencer and Lindzen both consider the lags important because correlation is greatly improved (and determines whether feedback is positive to negative). Neither seem to have mentioned the 3 month phase relationships emerging from integral/derivative system dynamics.

I can’t see how it is possible perform a valid analysis without this insight.

Here is the code.

figure0<-function() {
plot(x,x1,ylab="Magnitude",ylim=c(-2,2),lwd=5,xlab="Radians",main="Derivative and Integral of an Impulse",type="l")

figure3<-function() {

figure3.1<-function(X,data,lag=10,xlim=10) {
plot(c(-100,100),c(0,0),xlim=c(-xlim,xlim),ylim=c(-0.5,0.5),type="l",xlab="Years",ylab="Correlation",main="Cross-correlation of SW and LW with Global Temperature")
for (i in 1:dim(data)[2]) {

FFT of TSI and Global Temperature

This is the application of the work-in-progress Fast Fourier Transform algorithm by Bart coded in R on the total solar irradiance (TSI via Lean 2000) and global temperature (HadCRU). The results show (PDF) that the atmosphere is sufficiently sensitive to variations in solar insolation for these to cause recent (post 1950) warming and paleowarming.

The mechanism, suggested by the basic energy balance model, but confirmed by the plots below, is accumulation. That is, global temperature is not only a function of the magnitude of solar anomaly, but also its duration. Small but persistent insolation above the solar constant can change global temperature over extended periods. Changes in temperature are proportional to the integral of insolation anomaly, not to insolation itself.

The figure below is the smoothed impulse response resulting from the Fourier analysis using TSI and GT. This is the simulated result of a single spike increase in insolation. The result is a constant change, or step in the GT. This is indicative of a system that ‘remembers shocks’, such as a ‘random walk’. Because of this memory, changes in TSI are accumulated. (Not sure why its negative.)

Below is the Bode plot of the TSP and GT data (still working on this). The magnitude response shows a negative, straight trend, indicative of an accumulation amplifier. This is also consistent with the spectral plots of temperature that cover paleo timescales in Figure 3 here.

Bart’s analysis is going to be very useful doing this sort of dynamic systems analysis in a very general way. Up to now I have been using spectral plots and ARMA models.

This analysis above is an indication of the robustness of the method, as it gives a different but appropriate result on a different data set. Its going to be a very useful tool in arguing that the climate system is not at all like its made out to be.

I will post the code when its further along.

Global Atmospheric Trends: Dessler, Spencer & Braswell

Starting the S&B story at the beginning, as did Steve McIntyre, with Dessler 2010 in Science, I’ll put a new spin on the satellite data uploaded by Steve, using the accumulation theory. Although I am not familiar with the data, it turns out to be easily interpretable.

In black is the replication of Steve’s Figure 1 and Dessler’s 2010 Figure 2A, the scatter plot of monthly average values of ∆R_cloud (eradr) versus ∆T_s (erats) using CERES and ECMWF interim data. There is extremely little correlation as noted by Steve. In fact, it is not statistically significant in the conventional sense, Science apparently adopting the new IPCC-speak qualitative standard of ‘likely’.


Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.01751 0.04599 0.381 0.704
X 0.54351 0.36184 1.502 0.136

Residual standard error: 0.5036 on 118 degrees of freedom Multiple R-squared: 0.01876, Adjusted R-squared: 0.01045 F-statistic: 2.256 on 1 and 118 DF, p-value: 0.1358

The points in red are the sequential difference of temperature against the cloud radiance. While these have a lower slope, unlike the former, they are conventionally significant, almost to the 99%CL.


Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.01269 0.04524 0.280 0.7796
dX 1.07071 0.42782 2.503 0.0137 * ---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.4954 on 118 degrees of freedom Multiple R-squared: 0.0504, Adjusted R-squared: 0.04236 F-statistic: 6.263 on 1 and 118 DF, p-value: 0.01369

So why plot the sequential temperature differences and not the temperature directly? Firstly, while the autoregression coefficient (AR) of atmospheric temperature, erats (using arima(dess[,5],order=c(1,0,0))), is AR=0.65, for eradr its AR=0.16. This tells you that the two are different types of processes. The low AR is like a bunch of random numbers. The high AR is like a sequential accumulation of random numbers. Using different terminology, they do not cointegrate, as one can trend strongly (non-stationary) and the other stays around its mean (is stationary). Nor do they necessarily correlate. Though they can causally determine each other.

Differencing temperature is explained in accumulation theory, which pays close attention to heat accumulating in the ocean. Overlooking this basic physical model of the system causes many problems. Interpreting the data in terms of the physical model clears a lot of things up, as shown by the significant result above.

Above is the time-series plot of cloud radiance (black) and differenced global temperature (red) showing the relationship.

What does this say about cloud dynamics? The way to get intuition of dynamic relationships is to imagine the output from three types of input: impulse, step and periodic.


On an impulse of radiation, the surface (and lower atmosphere) warm and then revert. The differenced variable (like the first derivative) surges positive while temperature is rising, then surges negative while temperature is falling.

Electrical engineering buffs will appreciate this as the current-opposing behavior of an inductor. Clouds, in this view, could be compared with the electromagnetic field set up by the changing current. (The ocean heat capacity is comparable to capacitance).

The peak of the differenced pulse will lead the peak of the forcing. This shows it that lag/lead relationships are not reliable indicators of the direction of causation in dynamic systems.


On a step increase in radiation, the surface (and lower atmosphere) will ramp up as long as the forcing persists in accumulating heat in the ocean. The differenced variable will step-up and remain constant while temperature is rising at a constant rate.

This is a fundamentally different view of climate sensitivity, with different units. From the results above, the positive feedback from clouds is 1.1 W/m^2/K^2 and not 0.54 W/m^2/K. This means that clouds provide back-radiation (feedback is positive) while temperature is rising, but when the temperature stops rising, the back-radiation stops too. The number is part of the process.

I do not see how it is possible to interpret this in terms of a particular climate sensitivity. In the alternative view, cloud feedback is twice as strong as the conventional view while temperature is rising, but drops to zero when temperature is stable.


Finally, a periodic forcing is phase shifted 90 degrees (as shown by the impulse example). By simple calculus, the derivative of a sine function is a cosine function.

Could this explain the approximately 4 month lag in terms of an annual cycle (12/4 = 3 months)? Possible? It may explain the negative correlation achieved by Steve McIntyre at 4 month lag, as a 90 degree lead in a peak, produces a 90 degree lag between peak and trough.

Here is my code (you need to download the data from link above).


figure1<-function(X,Y) {
plot(0,0,cex=1,col=2,xlab="Global Temp (black) and diffTemp (red)",ylab="Clouds R",type="p",xlim=c(-0.4,0.4),ylim=c(-2,2))
plot(Y,col=1,ylab="Cloud R and diff(Temp)")


Phase Plots of Global Temperature after Eruptions

Here are a few more phase plots of global temperature after the impulse of stratosphere-reaching eruptions, Mt Agung, Mt Chichon and Mt Pinatubo in 1963, 1982 and 1991 respectively. The impulses are cooling of course, due to the shielding of short-wave solar radiation by stratospheric aerosols. The tendency of the global temperature dynamic to oscillate around a mean is clear.

These patterns were then disrupted by large El Ninos.

The axes of the phase space are chosen to represent abstract position and momentum (in this case temperature and temperature changes). Position and momentum in a conserved system correspond to potential and kinetic energy. The appearance of a circle or a spiral is evidence of a system that conserves energy by transferring between potential (radiative imbalance in this case) and kinetic (mass transfer, convection?) so that the sum remains constant.

Phase Lag of Global Temperature

Lag or phase relationships are to me one of the most convincing pieces of evidence for the accumulative theory.

The solar cycle varies over 11 years on average like a sine wave. This property can be used to probe contribution of total solar insolation (TSI) to global temperature.

Above is a plot of two linear regression models of the HadCRU global temperature series since 1950. The time since 1950 is chosen because it is the period that the IPCC states that most of the warming has been caused by greenhouse gasses GHG, like CO2, and because the data is more accurate.

The red model is a linear regression using TSI and a straight line representing the contributions of GHGs. This could be called the conventional IPCC model. The green model is the accumulated TSI only, the model I am exploring. Accumulative TSI is calculated by integrating the deviations from the long-term mean value of TSI.

You can see that both models are indistinguishable by their R2 values (CumTSI is slightly better than GHG+TSI at R2=0.73 and 0.71 respectively).

You can also see a lag or shift in the phase of the TSI between the direct solar influence (in the red model) and the accumulated TSI (green model). This shift comes about because integration shifts a periodic like a sine wave by 90 degrees.

While there is nothing to distinguish between the models on fit alone, the shift provides independent confirmation of the accumulative theory. Volcanic eruptions in the latter part of the century obscure the phase relation over this period somewhat, so I look at the phase relationships over the whole period of the data since 1850.

Above is the cross-correlation of HacCRU and TSI (ccf in R) showing the correlation at all the shifts between -10 and +10 years. The red dashed line is at 2.75 years, a 90 degree shift of the solar cycle, or 11 years divided by 4. This is the shift expected if the relationship between global temperature and TSI is an accumulative one.

The peak of the cross-correlation lies at exactly 2.75 years!

This is not a result I thought of when I started working on the accumulation theory. The situation reminds me of the famous talk by Richard Feynmann on “Cargo Cult Science“.

When you have put a lot of ideas together to make an elaborate theory, you want to make sure, when explaining what it fits, that those things it fits are not just the things that gave you the idea for the theory; but that the finished theory makes something else come out right, in addition.

Direct solar irradiance is almost uncorrelated with global temperature partly due to the phase lag, and partly due to the accumulation dynamics. This is why previous studies have found little contribution from the Sun.

Accumulated solar irradiance, without recourse to GHGs, is highly correlated with global temperature, and recovers exactly the right phase lag.

Accumulation of TSI comes about simply from the accumulation of heat in the ocean, and also the land.

I think it is highly likely that previous studies have grossly underestimated the Sun’s contribution to climate change by incorrectly specifying the dynamic relationship between the Sun and global temperature.

Climate Sensitivity Reconsidered

The point of this post is to show a calculation by guest, Pochas, of the decay time that should be expected from the accumulation of heat in the mixed layer of the ocean.

I realized this prediction gives another test of the accumulation theory of climate change, that potentially explains high climate sensitivity to variations in solar forcing, without recourse to feedbacks, or greenhouse gasses, in more detail here and here.

The analysis is based on the most important parameter in all dynamic systems, called the time constant, Tau. Tau quantifies two aspects of the dynamics:

1. The time taken for an impulse forcing of the system, such as a sudden spike in solar radiation, to decay to 63% of the original response.

2. The inherent gain, or amplification. That is if the Tau=10, the amplification of a step increase in forcing will be x10. This is because at Tau=10, around one tenth of an increase above the equilibrium level will be released per time period. So the new equilibrium level must be 10 times higher than the forcing, before the energy output equals the energy input.

I previously estimated Tau from global temperature series, simply from the correlation between successive temperature values, a. The Tau is then given by:

Tau = 1/(1-a)

Pochas posted the theoretical estimate of the time constant, Tau, below, that results from a reasonable assumption of the ocean mixed zone depth of 100m.

The input – output = accumulation equation is:

q sin ωt /4 – kT = nCp dT/dt

where q = input flux signal amplitude, watts/(m^2 sec). The factor 4 corrects for the disk to sphere surface geometry.

k = relates thermal flux to temperature (see below) J/(sec m^2 ºK).

T = ocean temperature,

ºKn = mass of ocean, grams.

Cp = ocean heat capacity J/(g ºK)t = time, sec or years.

Rearranging to standard form (terms with T on the left side):

nCp dT/dt + kT = q sin ωt /4

Divide by k

nCp/k dT/dt + θ = q sin ωt /(4k)

The factor nCp/k has units of time and is the time constant Tau in the solution via Laplace Transform of the above.

n = mass of water 100 m deep and 1m^2 surface area = 10E8 grams.

Cp = Joules to heat 1 gram of water by 1ºK = 4.187 J/gram.

k = thermal flux equivalent to blackbody temperature, J/(m^2 sec ºK).

Solution after inverse transform, after transients die out:

Amplitude Ratio = 1/(1+ω²T²)^½

where ω = frequency, rad/yr

Derivation of k Stefan Boltzmann equation

q = σT^4k = dq/dt

Differentiating: dq/dt = 4σT^3

Evaluating at T = blackbody temp of the earth, -18 ºC = 256 ºK

k = 4 (5.67E-8) 256^3 = 3.8 J/(sec m^2 ºK)

Calculating Time Constant Tau

Tau = nCp/k = 10E8 (4.187) / 3.8 = 1.10E8 sec

Tau = 1.10E8 / 31,557,000 sec/yr = 3.4857 yr


The figure of Tau=3.5 yrs is in good agreement with the empirical figures from the correlation of the actual global surface temperature data of 6 to 10. The effective mixed zone may be closer to 150m, and so explains the difference.

This confirms another prediction of the theory that amplification of solar forcing can be explained entirely by the accumulation of heat, without recourse to feedbacks from changing concentrations of greenhouse gases.

Solar Supersensitivity – a new theory?

Do the results described here and here constitute a new theory? What is the relationship to the AGW theory? What is a theory anyway?

The models I have been exploring, dubbed solar supersensitivity, predict a lot of global temperature observations: the dynamics of recent and paleoclimate climate variations, the range of glacial/interglacial transitions, the recent warming coinciding with the Grand Solar Maximum, and the more recent flattening of warming.

They make sense of the statistical character of the global temperature time series as an ‘almost random walk’, the shift in phase between solar insolation and surface temperature, and the range of autoregressive structure of temperature series in the atmosphere. These are all dynamic phenomena.

Conventional global warming models, based in atmospheric radiative physics, explain static phenomena such as the magnitude of the greenhouse effect, and are used to estimate the equilibrium climate sensitivity. The climate models, however, have very large error bands around their dynamics, and describe shorter term dynamics as chaotic. Does this mean they are primarily theories of climate statics, and supersensitivity is concerned with dynamics?

No. I see no reason why the accumulation theory could not be reconciled with coupled ocean/atmosphere general circulation models, once the parameterisation of these models is corrected, particularly the gross exaggeration of ocean mixing. Similarly there is no reason a model based on the accumulation of solar anomaly could not recover equilibrium states.

The difference between AGW theory and solar supersensitivity (SS) might lie more in the mechanisms. SS treats the ocean as a conventional greenhouse — shortwave solar isolation is easily absorbed, but the release of heat by convection at the ocean/atmosphere boundary is suppressed, so gradually warming the interior. In contrast, conventional AGW theory is focused more on mechanisms in the atmosphere, the direct radiative effects of gasses and water vapor. It combines many theories, of CO2 cycling, water relations, meteorology.

If mechanisms differentiate the theories, then the issue is the relative balance of the two mechanisms. Which is more responsible for recent warming? Which is more responsible for paleoclimate variations?

From basic recurrence matrix theory, the system with the largest eigenvalue will dominate the long-term, ultimate dynamics of a system, suggesting the ocean-related low loss accumulative mechanisms would dominate the short time-scale, high loss, low sensitivity atmospheric mechanisms.

If this view is correct, then what we have is a completion of an incomplete theory that promises to increase understanding and improve prediction by collapsing the range of uncertainty in the current crop of climate models.

Solar Supersensitivity – a worked example

Below is a worked example of the theory of high solar sensitivity, supersensitivity if you will, explained in detail in manuscripts here and here.

The temperature increase of a body of water is:

T = Joules/(Specific Heat water x Mass)

The accumulation of 1 Watt per sq meter on a 100 metre column of water for one year gives an expected temperature increase of

T = 32 x 10^6/(4.2 x 10^8)

= 0.08 C

Given that about one third attenuation of radiation from top-of-atmosphere to the surface, and a duration of solar cycle of 11 years, the increase in temperature due to the solar cycle will be:

Ta = 0.08 x 11 x 0.3 = 0.26 C

The expectation of the temperature increase for the direct forcing (no accumulation) using the Plank relationship of 0.3C/W would be 0.09 C. So the gain is:

Gain = Accumulated/Direct = 0.26/(0.3×0.3) = 3

For a longer accumulation of solar anomaly, from a succession of strong solar cycles such as we saw late last century, the apparent amplification will be more. From the AR correlation of surface temperature you get an estimate of gain of 10. But this is only apparent amplification, as the system is accumulative, the calculated gain increases with the duration of the forcing. For long time scales, gain (and hence solar sensitivity) approaches infinity — a singularity — and ceases to be useful. Hence the term ‘supersensitivity’. For long periods the non-linearity of the Stephan-Boltzmann law will become dominant.

Sensitivity cannot be represented in Watts/K (or K/Watt). It will be in units of rate like K/Watt/Year.

Extend this calculation for 1000 years and a small solar forcing can cause a transition between ice ages with no other input. The role of GHGs, water vapor and albedo in this theory is to maintain the heat state of the system, e.g. solar forcing increases temperature increase which causes CO2 concentrations to change. But this does not mean an increase in CO2 ‘necessarily’ increases temperature, because the system is being heated by accumulation of solar anomaly. The reason that a forcing from CO2 has apparently very low sensitivity, but solar very high, would be due to other issues that I haven’t worked through fully yet (coming soon).

Global Warming Temperature Trends

Roy Spencer posted the following comparison between the 20th Century runs from most (15) of the IPCC AR4 climate models, and Levitus observations of ocean warming during 1955-1999. Here are the best 4 models:

The accuracy of the other models is far worse.

In Roy’s assessment:

Previous investigators (as well as the IPCC AR4 report) have claimed that warming of the oceans is “consistent with” anthropogenic forcing of the climate system.

The actual rate of accumulated heat — the area between the green dots and the vertical line — much smaller than any of the models.

As Roy notes, it is generally believed that all of the increase in ocean heat is from increasing GHGs.

It should be mentioned the above analysis assumes that there has been no significant natural source of warming during 1955-1999. If there has, then the diagnosed climate sensitivity would be even lower still.

As I show here, the observations are consistent with the accumulation of heat from an excess of 0.2W/m^2 forcing by the Sun over the period of the Grand Solar Maximum. In other words, the increase caused by GHGs may not even be detectable.

An increase of 0.1W/m2 for one year would move 3.1×10^6 Joules of heat (31×10^6 sec in a Yr) to the ocean, heating the mixed zone to 150m by 0.006K (at 4.2 J/gK), producing a rate of global temperature increase of 0.06K per decade.

Accumulation Theory of Solar Influence

The physical structure of the oceans and atmosphere entails very long equilibrium dynamics due the slow accumulation of heat in the land and ocean. An ARMA analysis evaluates the potential of accumulation of solar anomaly to explain the global temperature changes over glacial/interglacial and recent time-frames.

Click image above for animation of the accumulation model for the 1950-2011 period.

The results of an early version of the accumulation theory are here.

Contrary to the consensus view, the historic temperature data displays high sensitivity (x10 gain) to solar variations when related by slow equilibration dynamics. A variety of results suggest that inappropriate specification of the relationship between forcing and temperature may be responsible for previous studies finding low correlations of solar variation to temperature. The accumulation model is a feasible alternative mechanism for explaining both paleoclimatic temperature variability and contemporary warming without recourse to increases in heat-trapping gases produced by human activities.

There are no valid grounds to dismiss the potential domination of 20th century warming by solar variations.

UPDATE: David Hagen alerted me to a post at WUWT where sun-spots were accumulated from 1500.

The sunspot record needs to be examined in its entirety rather than as individual sunspot cycles. The method to do this is by calculating the accumulated departure from the average of all the sunspot numbers of the entire 500-year index. This reveals the cooling during the Maunder Minimum and the current “global warming”. The current warming of 15 watts per square meter began in 1935, based on the sunspot record.

ARIMA theory of climate change

I have just uploaded a manuscript to the preprint archive viXra. ViXra is an interesting alter-ego to the other preprint archive arXiv. The goals of viXra are:

It is inevitable that viXra will therefore contain e-prints that many scientists will consider clearly wrong and unscientific. However, it will also be a repository for new ideas that the scientific establishment is not currently willing to consider. Other perfectly conventional e-prints will be found here simply because the authors were not able to find a suitable endorser for the arXiv or because they prefer a more open system. It is our belief that anybody who considers themselves to have done scientific work should have the right to place it in an archive in order to communicate the idea to a wide public. They should also be allowed to stake their claim of priority in case the idea is recognised as important in the future.

A Calibrated Water Tank

The dynamics of a surge tank, used to suppress damaging over-pressure in fluid lines, is described by a simple ordinary differential equation (ODE) the same as eqn. 1 in the recent paper by Spencer and Brasswell, “On the Misdiagnosis of Surface Temperature Feedbacks from Variations in Earth’s Radiant Energy Balance”, so very generally applicable to many systems.

The dynamics has been coded into an Excel spreadsheet by ControlsWiki, with an example of default variables above. Note:

1. The variability of the height of fluid in the tank (red) is suppressed and lagged, but there is no parameter in the model for lag. Lag is an emergent property of the system.

2. Increasing the periodicity of the flow into the tank increases the mean height of the water in the tank (try it). So the average of the water height is not directly related to the average inflow, but also depends on the period, another surprising emergent property.

This simple example shows how simple dynamics can lead to surprising behavior, and why misdiagnosis occurs by failing to account for dynamics when applying a simple linear regression, say. The first step in a correct analysis of a system is a valid physical model.

You can change the parameters and experiment with other aspects of the model.