The equation is presented in the pdf made public
3/10/11 at this 1 website. It is repeated here for convenience.
anom(Y) = calculated temperature anomaly in year Y
N(i) = average daily Brussels International sunspot number in year i
Y = number of years that have passed since 1700 (or any other year where the net summation is approximately zero such as 1856, 1902, 1910, 1938, or 1943)
T(i) = agt (average global temperature) of year i in °K,
ESST(c,Y) = ESST (Effective Sea Surface Temperature) in year Y calculated using an ESST range (magnitude) of c
CO2(Y) = ppmv CO2 in year Y
CO2start = ppmv CO2 in 1880
a, b, c, and d are coefficients to be determined.
Note that the energy gain from the sun is appropriately reduced by energy lost by radiation from the planet. The coefficient ‘b’ is the effective thermal capacitance. It relates the net energy in the numerator to a temperature anomaly.
The equation posits that average global temperature (agt) can be calculated from (1) the timeintegral of sunspot numbers (a proxy that correlates with energy retained by the planet), (2)predefined effective sea surface temperature (ESST) and (3) the measured atmospheric carbon dioxide (CO2) level. The influence of CO2 can be zeroed out by setting the coefficient ‘d’ to zero.
The projections made and graphed in the 3/10/11 pdf used the coefficients that were determined by a best fit of calculated temperature anomalies from 1895 to 2010 to the average of the
measurements reported by NOAA, GISS and Hadley Center. The equation with these coefficients was used to project the agt trend to 2037 as shown in the graph there.
The predictions are for two conditions:
1. Including the possible contribution from change to the level of atmospheric CO2.
2. Assuming that change to the level of atmospheric CO2 has no influence.
The infinitesimal (0.6%) change in R2 that occurs when change to the CO2 level is assumed to have influence of anywhere from none to 23.9% (the level at which R2 is maximum) of the calculated temperature rise from 1909 to 2005 indicates that change to the atmospheric CO2 level has no significant effect on agt.
Since the coefficients were determined using all available data, some reviewers asserted that the equation may have no predictive ability in spite of it being formulated from relevant physical
phenomena and a known law of thermodynamics.
To test the predictive ability of the equation, coefficients were determined for measurements through 1990 and these coefficients were then used in the equation to calculate predicted temperatures since then. Actual measured sunspot numbers and CO2 levels were used.
Calculations were made for the assumption that CO2 had influence and also the assumption that CO2 had no influence. Accuracy as indicated by the coefficient of determination, R2, increased only a tiny amount (0.84%) with increase in assumed influence of atmospheric CO2 from 0% to 21.6% (the level at which R2 is absolute maximum for measurements up to 1990) of the total calculated temperature rise from 1909 to 2005. This tiny change in R2 demonstrates that agt is insensitive to change in the amount of CO2 in the atmosphere. The results are shown in the
As this graph shows, the equation with either assumption of CO2 influence did a credible job of predicting the measured agt for the 20 years from 1990 to 2010. The equation using the oefficients determined assuming that added atmospheric CO2 has no influence did slightly better, especially after about 2005.
The barely detectable differences between the above graph, which is based on measured data through 1990, and the similar graph in the 3/10/11 pdf, which used measured data through 2010,
demonstrates that the equation has high predictive ability.
The coefficients that result from applying measured data from 1895 until the time noted are shown in the following table. The % rise due to CO2 is of the total calculated temperature rise
from 1909 to 2005.
The use of ESST smoothes the measured temperatures and/or measured ocean energy content which has been determined using temperature measurements. The indicated rapid year-to-year temperature changes as shown on the graph above are clearly impossible in light of the huge thermal capacitance of the oceans. The fluctuations are instead artifacts of the measurement process and are primarily indicative of the temperature non-homogeneity of the oceans.
The adjacent graph shows the agt response to a step change in the rate at which energy is retained by the planet. The effective thermal capacitance of the oceans has been variously determined by others 2,3,4 to be equivalent to from about 30 meters to 200 meters of depth.
This graph used an effective thermal capacitance of equivalent to 110 meters of depth. With this assumption, it takes about 4.5
years for agt to change by 70% of the total change (irrespective of the magnitude of the total change).
The oceans must be in constant slow upwelling and down welling. Enhanced evaporation at one location due to sun and/or wind would result in local decrease in temperature and increase in salinity both of which increase density leading to down welling with attendant upwelling elsewhere. Rain at one location would reduce local salinity and therefore density causing these areas to rise and flow away. Evidence of ocean surface temperature fluctuation is apparent in animations 5,6,7,8.
The temperature variations are tiny and produce the fluctuations on a timescale of a year or so at the points of measure as reported. The significant understanding of this is that the indicated fluctuations are artifacts of measurement. On the time scale of a year or two the true average temperature of the oceans changes very little.
The average trend of these measurements is another matter. The trend indicates a fluctuation in ESST of +/- about 1/6 °C with alternating up-trends and downtrends lasting about 32 years each.
The most recent trend high point was in 2005 and the most recent trend low point was in 1973.
The equation combines ESST trends with the energy retained by the planet as indicated by the time-integral of sunspot numbers and calculates temperatures since 1895 with an accuracy of 88%.
Comparison with the assumption that only CO2 has influence on agt
The equation can also produce a temperature anomaly trajectory for the assumed condition that only CO2 has influence on agt.
This is accomplished by simply setting ‘b’ to a very large number, ‘c’ to zero and adjusting ‘a’ and ‘d’ to get the maximum possible R2. The values thus derived for ‘a’ and ‘d’ as shown in the table above are then applied to project future temperature anomalies.
The temperature anomaly trajectory assuming that only CO2 has influence is shown on the next graph along with the temperature anomaly trajectory assuming that CO2 has no influence but ESST and the sunspot time-integral do have influence.
Future temperature measurements will quickly demonstrate the more correct of these two assumptions.
This equation predicts a future agt downtrend irrespective of whether it is assumed that atmospheric CO2 has an influence or not. As shown, the prediction that was made with the assumption that CO2 has no significant influence provides a better fit for the period after 2005.
The projection from 2011 that this equation makes, assumes that the sunspot time-integral is like it was from 1915 to 1941, that ESST trends continue like they have for over a century and that
atmospheric CO2 change has no influence.
The projection is a decline trend of about 0.13°C per decade. The decline trend will be steeper (up to about 0.22°C per decade) if the sun goes really
The accuracy of prediction of agt depends on the accuracy of prediction of sunspot numbers.
The calculated influence of CO2 on agt is between zero and about 25%. Expected continuation of the on-going declining trend in measured temperatures will result in an attendant decline in the
upper limit of this range.
The graph in the pdf made public 3/10/11 that shows temperature measurements since 1998 by the five reporting agencies and their average is updated as follows: A line showing the expected decline slope of 0.13°C per decade is added.
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