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() {
x=2*pi*seq(-1,1,by=0.01);x2=2*pi*seq(-0.5,0.5,by=0.01)
x1=c(rep(0,50),cos(x2),rep(0,50))
png("impulse.png");
dx=as.numeric(scale(c(0,diff(x1))));sx=as.numeric(scale(cumsum(x1)))
plot(x,x1,ylab="Magnitude",ylim=c(-2,2),lwd=5,xlab="Radians",main="Derivative and Integral of an Impulse",type="l")
lines(c(-2*pi/4,-2*pi/4),c(-2,2),col="gray",lty=2)
lines(c(0,0),c(-2,2),col="gray",lty=2)
lines(c(2*pi/4,2*pi/4),c(-2,2),col="gray",lty=2)
lines(x,sx,col=2,lwd=3)
lines(x,dx,col=3,lwd=3)
text(c(-2*pi/4,0),c(1.5,1.5,1.5),c("f'(t)","f(t)=cos(t)"))
text(2*pi/4,1.5,expression(paste("u222B",f(t))))
dev.off()
browser()
png("cross.png");
cxd=ccf(dx,x1,lag.max=100,plot=F)
cxs=ccf(sx,x1,lag.max=100,plot=F,new=T)
w=2*pi*cxd$lag/(100)
plot(w,cxs$acf,col=2,type="h",xlab="Radians",ylab="Correlation")
lines(w,cxd$acf,col=3,type="h")
lines(c(-100,100),c(0.15,0.15),lty=2,col=4)
lines(c(-100,100),c(-0.15,-0.15),lty=2,col=4)
lines(c(-100,100),c(0,0))
dev.off()
}

figure3<-function() {
par(mfcol=c(1,1),mar=c(4,4,3,3))
figure3.1(spencer[,7],spencer[,1:6],xlim=1)
#par(mar=c(4,4,0,3))
#figure3.1(dess[,5],dess[,1:4],xlim=1)
}

figure3.1<-function(X,data,lag=10,xlim=10) {
png("impulse.png");
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")
lines(c(-100,100),c(0.18,0.18),lty=2,col=4)
lines(c(-100,100),c(-0.18,-0.18),lty=2,col=4)
lines(c(0.25,0.25),c(-1,1),lty=3)
lines(c(-0.25,-0.25),c(-1,1),lty=3)
send=tsp(data)
labels=colnames(data)
t=window(X,start=send[1],end=send[2])
for (i in 1:dim(data)[2]) {
cxd=ccf(data[,i],t,lag.max=lag,plot=F)
w=cxd$lag
lines(w,cxd$acf,col=i+1,lwd=2)
text(0.9,cxd$acf[length(w)],labels[i],col=1,cex=0.5)
}
dev.off()
}

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’.

Coefficients:

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.

Coefficients:

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.

Impulse

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.

Step

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.

Periodic

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).

dess=ts(read.csv(file="dessler_2010.csv")[3:8],start=2000.167,frequency=12)

figure1<-function(X,Y) {
dX=ts(c(0,diff(X)),start=start(X)[1],frequency=frequency(X))
fm=lm(Y~X)
fm2=lm(Y~dX)
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))
points(X,Y,col=2)
abline(fm,col=2)
points(dX,Y,cex=1,col=1)
abline(fm2,col=1)
browser()
plot(Y,col=1,ylab="Cloud R and diff(Temp)")
lines(dX,col=2)
lines(X,col=3)
}

figure1(dess[,5],dess[,3])

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.