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()
}

35 Comments

  1. Stephen Wilde September 21, 2011 11:04 pm

    You said:

    “1. Changes in cloud cover actually do drive changes in global temperature due to gamma-ray flux (GRF) or other effects, o
    2. The changes in cloud cover are caused by changes in global temperature, with the derivative mechanism described above.
    3. Both 1 and 2″

    There is another option which I prefer in light of real world observations.

    A solar induced change in the vertical temperature profile of the atmosphere shifts the surface air pressure distribution latitudinally so as to change global cloud quantities.

    That alters solar shortwave input to the oceans thereby affecting temperatures.

    Would that fit your phase shift data?

    • Anonymous September 22, 2011 2:31 am

      Yes. Wouldnt that be covered by 1?

      • Stephen Wilde September 22, 2011 6:29 am

        Broadly yes, but really it is a mix of 1 and 2 amounting to 3 because the warmer temperatures then affect cloudiness too so neither 1 or 2 on their own are sufficient.

        My main concern is as to whether that specific concept would sit well with the rest of the article and in particular the equations.

        I agree with Spencer that it would be impossible, currently, to separate the top down effect on cloudiness from the solar variations from the bottom up effect on cloudiness from oceanic variations.

        However the net outturn would clearly be a consequence of the interaction between both top down and bottom up processes.

        If we could calculate the total cloudiness at any given time and monitor changes as they occur then that should give us an indication aa to when changes in global temperature trend occur and the speed of such changes and in due course maybe a figure for global cloud quantity at which the system would be approximately in thermal balance.

        • Davids99us September 22, 2011 8:08 am

          The clarity offered by control theory covers these problems, if you have been following http://climateaudit.org/2011/09/08/more-on-dessler-2010/ you get the idea.  I think its really hard to talk about these systems without the math.

          But the main point of the first figure is to illustrate that the peak intensity of an effect can occur before the cause.  This doesnt reverse cause and effect.  its just because the peaks is not a good indication of the direction of causation.

          In fact, I don’t like to think about causation at all. I try to think in terms of equations satisfying conservation equations. 

          • David L. Hagen September 22, 2011 7:31 pm

            David s.
            The derivative and integral have the opposite sign. Can these be used to distinguish between your options 1 and 2?
            See WebHubTelescope being able to use a Proportional Derivative model to obtain an amazing fit of dCO2 and T with time.

            Could these difference in phase, sign, or lag, or between the proportional and derivative be used to further distinguish between solar cloud modulation with consequent CO2 ocean changes, vs Fossil CO2 emissions changing the optical depth causing ocean temperature changes?

          • Davids99us September 23, 2011 12:07 am

            It is interesting and highly unlikely that the dCO2 would affect temperature, but temperature proportionate to dCO2 must be an expression of OD equations. 

            Im not familiar enough with the data to be 100% sure if the difference in signs is due to lags, leads, or different polarity.

  2. David L. Hagen September 24, 2011 8:22 pm

    Fred
    Haynie’s Slide 16/59
    shows about a 3 month or 9 month lag between Arctic
    and Antarctic ice extremes. That looks reasonably close to the 0.25 year and
    0.75 year lags (0.25 + 0.5) between insolation and temperature minimums predicted
    by Stockwell’s solar accumulation theory. Correspondingly, in slide 20/59, the Sea
    Ice thaw rates (as proxy for heat rates) look to be in synch with the respective north and southern pole insolations.
    See urther comments on CO2 lag similar to temperature lag.

  3. Curt September 28, 2011 9:20 pm

    Speaking as someone who has worked professionally using systems analysis for 25 years now (and who sometimes teaches it at the university level), I do not consider the attribution of the lag to “thermal inertia” to be incorrect, because it automatically implies an integrative effect between power input and resulting temperature.

    As you say, the phase lag to a cyclic input (at least after the startup response decays away) does not depend on the magnitude of the “thermal inertia”, at least for lossless integration, but the magnitude of the response is inversely proportional to this value. And the lag to non-steady-state and/or non-cyclic forcings does depend on the magnitude.

    Of course, if there are “losses” — e.g. convection or radiation transfers that vary with temperature, you do not get the pure 1/4-cycle lag at all frequencies, but a more complex relationship.
    Surface air temperature response to diurnal and annual forcings has closer to a 1/8-cycle lag.

    • Anonymous September 28, 2011 10:13 pm

      Curt, All correct. But you get my point about the misconception. Systems analysts understand the issues in unravelling climate science very well, but it seems climate scientists have very little clue in this area and either linear regress things or accept climate model outputs blindly. Systems analysis could contribute a lot.

      • Anonymous September 29, 2011 3:39 am

        “Systems analysis could contribute a lot.”

        Only if they publish results that support AGW.  Otherwise their results must be wrong somewhere.  Faulty observation, bad math, mistaken assumptions.  Or paid shills of the oil industry.

        • Davids99us September 29, 2011 5:55 am

          I sometimes wish I had the emotional intelligence to handle that 5 years ago.

      • Curt September 29, 2011 2:32 pm

        Yeah, I suspect we’re pretty much together on this, except possibly for the semantics. I have long cringed at the way the basic block diagram for climate is presented, as a “pure gain” term, with no dynamics — completely unphysical, and not likely to lead to meaningful insights (which should be the main purpose of these simple models).

        At a minimum, the basic model should be that of an integrating thermal capacitance (I like to use the electrical analogy term) with a loss term fed back around it. As a first cut, this loss term can be linearized, at least for small perturbations. From this two-element model, you can derive equilibrium sensitivity, time constant, phase shift at various frequencies, etc.