How global warming works

For Americans who don't mind seeing a few numbers

Vaughan R. Pratt

Stanford University


This account of global warming is for Americans who don't mind seeing a few numbers. Americans because they prefer feet and Fahrenheit to meters and Celsius, and a few numbers for those who like a little perspective. If you see a centimeter in this account feel free to step on it.

Everyone uses the same units of seconds and years for time. As for weight, a long ton, or imperial ton, is 2240 lbs while a metric tonne is 2206 lbs, about 1.6% less. I'll give weights in tons which you can assume is close enough to both of those scales for our purposes.

A metric tonne is exactly a million grams or one megagram, so a trillion tonnes (a million million tonnes) is a trillion million grams or one exagram, abbreviated 1 Eg. This is the most convenient unit for quantifying Earth's atmosphere, which weighs 5,150 Eg.

For comparison this is the same weight as the top 47 feet of the water in the ocean. However if that weight of water were somehow spread over the whole of Earth's surface it would only be 33 feet deep. This gives us another unit of weight for the atmosphere: equivalent feet of water.

Whereas the mass of the whole atmosphere is 33 feet in that scale, the mass of its greenhouse gases back in 1800 was about 1.15 inches, of which about 15% by weight was CO2. Since then CO2 has risen to 20% of greenhouse gases and at the present rate of increase will be up to 36% by 2100. The effect of greenhouse gases will be an important part of this account of modern climate.

What warms Earth's surface?

The Sun, of course.

The Sun warms the Earth across 93 million miles of space

If Earth retained all that incoming heat it would keep getting hotter until the oceans boiled away.

Earth's oceans all boiled away

Obviously that hasn't happened.

This is because as Earth gets hotter it radiates more of that incoming heat to space.

Outgoing Longwave Radiation (far infrared wavelengths)

Once Earth has reached a temperature where the heat it is radiating to space exactly equals the heat it has been absorbing from the Sun, its temperature stops rising. This situation is called thermal equilibrium.

For any planet this temperature is called its effective temperature.

By pure coincidence, the effective temperature of Earth happens to be almost exactly 0 °F. (In Celsius or Centigrade, -18 °C, in Kelvin or absolute temperature, 255 K.)

Now fresh water freezes at 32 °F. Seawater freezes a little colder, namely at 28.4 °F. But 0 °F is way colder than that, and at that temperature the oceans should be frozen solid.

Earth's oceans all frozen solid

Obviously that hasn't happened either. For the most part the oceans are neither steam nor ice. So what's going on?

What's going on is that there is more to Earth than just its oceans and land. There is also its atmosphere.

For the purpose of radiating the Sun's incoming heat back to space, Earth's atmosphere does more than 90% of of that work. Earth's surface itself radiates a little to space, but the greatest part is radiated from the troposphere, constituting about three-quarters of Earth's atmosphere by mass. The troposphere ends at the tropopause, which is nominally at about 35,000 feet (but lower at higher latitudes). Above the tropopause is the stratosphere, which contains most of the remaining quarter of the atmosphere.

Lapse rate

As you rise above sea level, whether by balloon ride, in a plane, or climbing a mountain, you will find the air cooling at a rate of about 3.5 °F per thousand feet. This rate of cooling with altitude is called the atmosphere's lapse rate. By the time the balloon has reached 35,000 feet the temperature should have dropped about 120 degrees.

It turns out that the effective temperature of the Earth roughly coincides with the temperature at the middle of that 35,000 foot balloon ascent. Earth's radiation to space is holding the overall temperature of the atmosphere to ±60 °F on either side of that middle. So if at the middle of the ride the temperature is at 0 °F, then at the start of the ride it must be 60 °F and by 35,000 feet it must have fallen to -60 °F.

Now every point of the atmosphere radiates heat uniformly in all directions. That is, if you imagine a virtual sphere say a foot in diameter anywhere in the atmosphere, the radiation being emitted from the very center of that sphere is equally strong everywhere it passes through the surface of that sphere.

However the radiation directed downwards or sideways obviously can't escape to space and will be eventually captured. Only upwards radiation can escape to space provided it is not absorbed by the atmosphere on the way up. The steeper the upwards radiation the less atmosphere it has to punch through and the better its chances of escaping to space.

How can the atmosphere radiate?

About 99.7% of Earth's atmosphere by weight is completely transparent to Earth's thermal radiation, the kind of radiation that is needed to keep Earth's oceans from boiling away. Had it been 100%, all the radiation from Earth to space would be coming directly from the surface, whose temperature would therefore be Earth's effective temperature of 0 °F.

The remaining 0.3%, weighing about 15 Eg, consists of gases capable of both absorbing and radiating Earth's thermal radiation. Even though this is a very small proportion of the atmosphere it has nevertheless trapped and reradiated enough radiation that on average the radiation that finally makes it to space has been coming from about halfway up the 120 degree scale in the previous section. This altitude is therefore where the 0 °F temperature is. This makes Earth's surface temperature around 60 °F.

The altitude at which the radiation is emitted influences the amount that escapes to space in two ways. Hot gases radiate more strongly so emission at low altitudes is stronger. On the other hand the path to space is longer, reducing the chances of this stronger radiation escaping to space.

These two influences are summarized in the following picture.

OLR from low altitudes: stronger but less likely to reach space

The x-axis in this picture gives the altitude of emission, up to 35,000 feet. To avoid clutter it shows only the radiation directed vertically upwards.

And for further clarity the picture puts the radiation from the surface on the left and the radiation from 35,000 feet on the right, with all intermediate altitudes in between.

Radiation from low altitudes is stronger, shown with brighter lines, because of the higher temperature there. However it is also more likely to be captured before it can escape to space, shown with a greater proportion of lines stopping prematurely. Given this tradeoff it is not all that surprising that Earth's effective temperature of 0 °F occurs in the middle of that range.

But if it had occurred at only a quarter way up, that is at about 9,000 feet, then Earth's surface temperature would be only 30 °F. Since this is a couple of degrees below freezing, a lot more of Earth's oceans would be frozen!

What are the main greenhouse gases?

In terms of their warming potential the two main greenhouse gases are water vapor and carbon dioxide, CO2.

H2O and CO2 molecules each contain three atoms and their respective greenhouse effects can therefore be assumed to be roughly equal. Molecules like methane, CH4, have five or more atoms and because of this greater number their contribution to the greenhouse effect of each ton is considerably greater. However their mass is less than one percent of that of CO2 and their contribution to global warming is therefore negligible by comparison, their greater warming power per ton notwithstanding.

The mass of water vapor in the atmosphere is fairly constant at around 12.8 Eg. Before the industrial era CO2 contributed 2.2 Eg to that, bringing it up to 15.0 Eg. (To convert ppm of CO2 to its mass in exagrams, divide by 128, more precisely 127.84. For the mass of the carbon alone divide instead by 468.8. A commonly used unit is GtC for gigatons of carbon; 1 EgC is 1000 GtC.)

Back when the mass was 15.0 Eg, these two greenhouse gases added about 60 °F to the surface temperature. Each Eg of greenhouse gases can therefore be blamed for about 60/15 = 4 °F.

Since then however the mass of CO2 has increased by a further 1 Eg, bringing the total mass of the two main greenhouse gases to 16 Eg.

According to theory this increase in GHGs should raise the temperature by about 4 °F. But not necessarily right away as the oceans have an enormous thermal inertia that slows down the warming; it will take more than a century to reach thermal equilibrium, even assume no further increase in GHGs, maybe a lot more.

Now the following plot shows that land temperature has risen by about 2.3 °F over the past 50 years.

We can thank the sea breezes from the oceans for holding the rise in check.

If all of a sudden CO2 stops rising, in a few centuries that extra 1 Eg of greenhouse gases can be expected to increase the surface temperature by some 4 °F. Already we have seen the temperature increase by about 2 °F, more on land and less at sea. What is holding it back from the full 4 °F are the polar ice caps. These work like the ice blocks that the ice man used to bring around to keep ice boxes cold before the advent of the electric refrigerator. If those deliveries had ever ceased, in a few days your ice box would have warmed up to room temperature. For ice blocks the size of those at the poles, think centuries instead of days.

During the centuries while the ice is melting, that melt-ice is going to find its way into the oceans, which will therefore keep rising until equilibrium is reached. ("Equilibrium" is the scientific term for when things have settled down.)

But what are the chances of CO2 stopping rising today? To answer this we need to look at CO2 in excess of the natural background level of 280 ppm, which we can call anthropogenic CO2 or ACO2.

During the past half century ACO2 would appear to be rising with a compound annual growth rate or CAGR of about 2%, with no sign at all of any change. In 2014 that excess was 120 ppm or about 0.94 Eg. Over 80 years that 2% CAGR amounts to a factor of 5. So unless that CAGR starts to reduce, by 2094 ACO2 will have increased by an additional 4.7 Eg, and nearly 5 Eg by 2100!

So if in 2100 ACO2 stopped at 5 Eg more than in 2014, and somehowever remained there for the rest of the millennium with no further increase, this is 6 Eg more than since preindustrial times. In the long run, many centuries later, much ice would have melted and the temperature would have slowly risen to a new equilibrium some 6*4 = 24 °F above preindustrial times. During that period Earth's inhabitants would see the seas rising until all the ice at the poles had melted.

You and I won't be around even at the start of that period, so why should we care?

This is one of those difficult existential questions. If during that millennium Earth is hit by a bigger asteroid than any experienced in the last billion years, maybe it wouldn't matter at all, even to our descendants.

Likewise if you think your crazy hot-rodding son is more likely to die in a car accident than of lung cancer, what would be the point of bugging him about his unhealthy chain-smoking habit?

These are not questions my training in physics and logic has qualified me to answer. I rather imagine even ethicists can't answer that sort of question.

Where did you get 6 degrees from?

The 6 degrees comes from the following two things.

  • The last 170 years of rising CO2 and rising temperature. These permit an estimate of the dependence of temperature on CO2. If we define centennial climate as global mean surface temperature averaged over 65 years, then I estimate that each doubling of CO2 increases centennial climate by 3.3 °F.
  • The last 50 years of rising CO2. CO2 in excess of 280 ppm has been rising at an astonishingly steady rate of 2% a year. In 2014 it was at 400-280 = 120 ppm. Hence in 2100 or 86 years after 2014 it will be at 120*1.02^86 = 660 ppm. Adding 280 to that gives 940 ppm.
  • CO2 rising from 280 ppm to 940 ppm is an increase of a factor of 940/280 = 3.357. This is equivalent to log22(3.357) = 1.75 doublings of CO2. If each doubling raises temperature by 3.3 °F, the total rise during the industrial era will have been 3.3*1.75 = 5.8 °F.

    The Gathering Storm

    The history of greenhouse gases begins with the Swiss physician, mountaineer, and experimentalist Horace de Saussure in the 1780s who demonstrated that the Sun heated the tops of mountains just as strongly as the bottoms.

    It continued with French mathematician and theorist Joseph Fourier in the 1810s who made the connection with heat-trapping glass.

    Next came the Irish experimental physicist John Tyndall in the 1850s who showed that gases whose molecules had three or more atoms such as H2O and CO2 trapped infrared heat.

    In 1896 Swedish chemist Svante Arrhenius conducted an experiment to measure the amount of heat from the Moon trapped by the atmosphere as a function of the length of its path through the atmosphere, and inferred a climate sensitivity of 4 or 5 degrees C per doubling of CO2.

    In the 1930s British engineer Guy Callendar wrote extensively on the threat posed by growing CO2.

    In 1958 American scientist Charles Keeling oversaw the installation of a observatory to measure atmospheric CO2 at 9000 feet on Mauna Loa on Hawaii's Big Island.

    And in 1987 a handful of governments who'd been wondering about the three decades of Keeling's measurements asked the World Meteorological Organization and the United Nations Environmental Programme whether they needed to worry about rising CO2.

    In response the WMO and UNEP joined forces and in 1988 convened a number of scientists as the Intergovernmental Panel on Climate Change to summarize what was known to date about that possible threat. The IPCC did so in a report released in 1990 as its first Assessment Report.

    At the time the projections were primarily theoretical. The actual evidence for rising CO2 causing what the report's lead author referred to as global warming was very thin, as can be seen from the lack of correlation in this graph betwen Keeling's data on CO2 and the world's surface temperatures made on land and at sea.

    The only certain claim that could be made at the time was the obvious fact that, whatever was causing rising CO2 could not have been rising surface temperature, which was wobbling around in a way completely uncorrelated with the very steadily rising CO2.

    But every six years or so the IPCC would release another report, the Fifth Assessment Report or AR5 being released in 2013. By then surface temperature had started to rise. But then during 2002-2012 it came very close to plateauing, affectionately called The Hiatus and raising the possibility that surface temperature had risen about as far as it could and was about to return to its much colder 1970s levels.

    Fast forward to 2020. During the decade following 2012 it became clear that, far from being peak heat, the hiatus was merely a temporary pause, as can be seen from this 30-year extension of the foregoing plot to 1990.

    The right hand half of the previous graph, which at the time looked like merely part of a random wobble, can now be seen as the start of a long rise to today's much hotter climate.

    During 1958-2020, each 1 ppm rise in CO2 corresponds roughly to a 10 mK (millikelvin) rise in temperature. So a 1 °C rise in surface temperature corresponds to a 100 ppm rise in CO2, for example from 320 ppm to 420 ppm.

    These two graphs show Celsius rather than Fahrenheit degrees on the right axis. The range from -0.4 to 1 °C corresponds to -0.72 to 1.8 °F.

    Recent climate as defined by the World Meteorological Organization

    The blue curve in the following plot is HadCRUT4, global mean surface temperature (sea and land) since 1850.

    There would appear to be much randomness in this data, suggestive of chaotic influences in annual climate.

    The red curve is the same thing but smoothed to a running mean or moving average of 30 years. This is the World Meteorological Organization's definition of "climate". It has the benefit of eliminating the chaotic influences on climate so as to expose the general trend.

    This trend has two evident components, a long-term exponential growth on which is superimposed an oscillation with a period of some 60 to 70 years.

    Detrended WMO climate is WMO climate detrended by the expected influence of CO2 from WMO climate, as shown in this plot.

    1.5 CO2 between 1 CE (the year after 1 BC) and 1900 CE

    This plot shows how during the first 18 centuries of the Christian era CO2 did not rise above 286 ppm until about half a century after James Watt invented the efficient steam engine. From then on it climbed at an ever-increasing pace never seen before.
  • During the 19th century CO2 climbed 14 ppm, an unprecedented rise in any previous century of Earth's recorded history.
  • But then during the 20th century CO2 climbed an additional 71 ppm.
  • In just the first twenty years of this century CO2 has risen a further 43 ppm!
  • It is very clear that CO2 emissions are getting more and more out of hand.

    2. The CO2 "hockey stick" since 1000 CE

    The following two plots demonstrate that rising CO2 can be modeled to within 5 ppm as 280 ppm plus an exponential compounding annually at 2% (CAGR = 2%). The thin dark blue line in the first plot gives observed atmospheric CO2 as measured in Antarctic ice cores for pre-1958 atmosphere and more directly at the Mauna Loa CO2 observatory since 1958.
    The thick light blue curve is essentially the model of rising CO2 proposed in 2009 by David Hofmann, late of NOAA Boulder. It plots the function 280 + 1.0203y - 2014 ± 5 ppm.

    Now if we subtract 280, take the log base 2 of the exponential, and zoom in to the top right corner of the above graph, we obtain the following plot for the 60 years 1960-2019.

    As can be seen from the red and green trend lines respectively, ACO2 has been compounding annually at 2.43% for the first half of this period and 2.03% for the second half.

    And if we zoom in even closer, to just this millenium, and plot monthly rather than annual CO2, we observe the following remarkable fit of the 2.03% model to the observed level of rising CO2.

    3. Impact of CO2 emissions

    This graph for the period 1851-2019 plots greenhouse gases (the blue curve) and land temperature in degrees Fahrenheit (the red curve) as they rise from 15.1 to 16.1 Eg and from -1 °F to 2 °F respectively.
    The impression of another hockey stick is created by the decrease in temperature during 1940-1970.

    4. Attribution of detrended long-term climate to the Sun

    The two curves in the following graph show an excellent correlation between detrended long-term climate as represented by annual HadCRUT4 and long-term Sun as represented by a recent reconstruction of historical Total Solar Irradiance (TSI).

    After loading the data for annual HadCRUT4, CO2, and TSI, all for the period 1850-2017, MATLAB then plotted the blue and red curves with the following one-line MATLAB command. plot(1882:1985, [F(HadCRUT4 - 1.9*log2(CO2)) F(0.3*TSI)])

    A few additional lines annoted the graph and defined the function F(X) which smooths X to a 65-year moving average and then centers the result by subtracting its mean in order to align the curves vertically.

    The main points are as follows.

  • Each time series consists of 168 numbers, one for each year 1850-2017.
  • HadCRUT4 is global climate in degrees w.r.t. average over 1961-1990.
  • CO2 is ppm as measured from ice cores before 1958 and thereafter Mauna Loa.
  • TSI is total solar irradiance at top of atmosphere (TOA) in W/m2.
  • 1.9*log2 yields 1.9 degrees per doubling of CO2.
  • Subtracting it from HadCRUT4 is climate if CO2 had remained constant.
  • 0.3*TSI assumes that each W/m2 at TOA warms Earth by 0.3 °C.
  • The remarkable similarity of the two curves shows that the only long-term fluctuations in HadCRUT4 are due to CO2 and the Sun, with almost all from CO2.
  • If this behavior continues to 2100, the contribution of the Sun will remain within a range of ±0.03 °C. CO2 in excess of 280 ppm on the other hand has been compounding annually at 2% for two centuries now and if this keeps climate will be 3.4 °C above its preindustrial average.
  • This is only one possible explanation of what drives global climate and not a proof, as we have not ruled out the possibility of other completely different climate drivers matching global climate equally well.


    (Only the first few annotated)

    -4. Residual surface temperature compared with human radiative forcing

    Since 1800 global population has increased seven-fold, compounded by an even greater rise in per capita energy consumption.

    There is an evident correlation between the seven-fold rise in each of global population and technology over the past two centuries and the atmospheric CO2 "hockey stick", suggesting that humans are the cause of the latter, which in turn could be the cause of the one-degree rise in global climate over that period.

    A straightforward way to see the connection is by plotting rising atmospheric CO2 against emitted CO2 since 1900, as follows.

    This shows that each petagram (billion tons) of emitted CO2 has increased the mass of CO2 in the atmosphere by 0.56 petagrams. Equivalently, each PgC (petagram of carbon) has raised atmospheric carbon by 0.56 PgC--just multiply both figures by 12/44.

    The following plots the actual rise in temperature against the rise that would be expected if instead of 0.56 PgC we used 0.45 PgC, which unlike the 0.56 factor takes land use changes into account. The y-axis is HadCRUT4 filtered with an 11-year moving average filter to remove the solar cycle's influence and detrended by other natural influences.

    The graph plots RST against Human Radiative Forcing defined as follows.
  • ECO2 is emitted CO2 accumulated since 1900, expressed in units of ppm as per the Carbon Dioxide Information Analysis Center, CDIAC. The carbon in CO2 has mass 12/28.97 of air and the atmosphere has mass 5150 teratonnes, whence 1 ppm of atmosphere corresponds to 12/28.97*5150 = 2133 megatonnes of carbon (2.133 PgC, petagrams of carbon).
  • ECO2/280 changes the ppm units to multiples of preindustrial CO2. In these units preindustrial CO2 is 1, whence preindustrial radiative forcing equals log(1) which is zero.
  • The surface absorbs 55% of our emitted CO2, whence 0.45*ECO2/280 is the amount remaining in the atmosphere. Since surface absorption is reduced by deforestation and other land use changes, further such changes may increase the 0.45 fraction.
  • 0.45*ECO2/280 plus 1 (preindustrial CO2) is expected total atmospheric CO2 based on emitted CO2 since 1860.
  • log base 2 of that is radiative forcing in the case Climate Sensitivity CS equals 1. The graph shows that CS is 1.74. Note that this is neither Equilibrium Climate Sensitivity nor Transient Climate Response but rather the apparent instantaneous effect of the prevailing rate of rise of CO2, sometimes called Observed Climate Sensitivity.
  • One benefit of correlating temperature with emitted CO2 rather than measured atmospheric CO2 is that it makes natural sources of recently rising CO2 irrelevant to the question of whether humans are responsible for rising temperature. The above graph is strong evidence that they are.

    -2. CO2 vs. Methane

    The 20-year global warming potential of methane is 86, meaning that averaged over 20 years 1 part per million by mass has the same global warming potential as 86 parts per million by mass of CO2.

    Atmospheric concentrations are however given not by mass, ppmm, but by volume, ppmv. The respective masses of a mole of CO2 and a mole of CH4 are 44 and 16. while the mass of an average mole of atmosphere is 28.97. Hence 1 ppmm of CH4 is therefore 28.97/16 = 1.81 ppmv while 86 ppmm of CO2 is 86*28.97/44 = 56.6 ppmv. Hence when comparing CO2 and CH4 when their concentrations are given by volume, the global warming potential of CH4 is only 56.6/1.81 = 31 relative to CO2.

    Based on that factor of 31, the following graph compares the global warming potentials of CO2 and methane (CH4) since 1860. Each is given first as a model whose anthropogenic component is growing at 2% a year, and then its actual values.

    The most important difference is that CO2 is contributing much more to global warming than CH4, even when the anthropogenic components continuing to grow at 2%, and is showing no sign at all of slacking off, after 1990 ACH4 suddenly slowed down dramatically.

    What we need today is for ACO2 to slow down the way ACH4 did after 1990.

    There is one further difference between CO2 and CH4. Whereas CO2 does not decompose, CH4 decomposes into CO2 and H2O. The 100-year GWP of CH4 by mass is down to 25 while its 500-year GWP is a mere 7.6. By volume these two GWPs reduce to 9.1 and 2.8 respectively.

    -1. The main fingerprint of CO2 on global climate

    The following figure plots three curves.
  • What global climate since 1850 would have done even if CO2 had no impact at all.
  • What global climate since 1850 would have done if we had not increased CO2 the way we did.
  • What global climate since 1850 would have done if CO2 in excess of 280 ppm had not increased at 2% a year since 1850 so as that excess were to reach 120 ppm in 2015.
  • The first (blue) curve is HadCRUT4 itself, with the assumption that CO2 is not involved. If so, how would you explain a rise not seen for thousands of years?

    The second (red) curve is what HadCRUT4 would have recorded if CO2 had not rised as recorded.

    The third (green) curve is what HadCRUT4 would have recorded if CO2 had not rised as modeled.

    0. Direct observation of the Sun's influence on centennial climate

    We take climate to be global HadCRUT4 since 1850. By that definition climate is always changing, and furthermore very quickly.
    But even if we average climate over 30 years, the World Meteorological Organization's definition of climate (blue curve in this figure), climate is still changing up and down, though not so quickly.
    /center> If we average climate over 65 years however (red curve in that figure), climate is still changing, though only up, not down.

    However the Sun remains constant to within 0.1%.

    One would therefore not expect to be able to directly observe the influence of the Sun on climate. It is therefore surprising that its influence can be seen in the green and magenta curves in the middle of the following graph.

    The common scale for the first four curves is watts per square meter, W/m2. This graph tests the mutual consistency of four hypotheses.
  • An increase of 1 W/m2 of radiative forcing is exactly offset by an increase of 1 W/m2 of thermal radiation to space.
  • The radiative forcing attributable to a doubling of CO2 is 1 W/m2.
  • In the absence of feedbacks 1 W/m2 of radiative forcing raises surface temperature by 1 °C.
  • The net gain of all climate feedbacks is 0.46.
  • With that gain, each W/m2 of radiative forcing should raise the temperature by 1/(1 - .46) = 1.87 degrees. Since radiative forcing due to rising CO2 and solar irradiance are both given in W/m2, instead of converting both to temperatures we infer the radiation responsible for warming climate by multiplying HadCRUT4 by (1 - .46) = 0.54. We then detrend it by subtracting log2(CO2) as the hypothesized contribution of radiative forcing due to rising CO2. This is the blue curve labeled DeCl in the legend.

    The absorbed portion of TSI is (1 - A)*TSI, being the portion of TSI not reflected straight back to space as outgoing shortwave radiation. ASI is what heats Earth's atmosphere and surface. The red curve labeled QASI is one quarter of ASI and is the flux density of heat escaping to space, averaged over Earth's surface, that would exactly offset ASI. The factor of four is the ratio of Earth's surface (as a sphere of surface area 4πr²) to Earth's cross section (as a disc of area πr² as seen from the Sun).

    To align DeCl and QASI vertically we center both by subtracting their respective means.

    The evident fluctuations in both DeCl and QASI make it impossible to see any relationship between them. We can eliminate the faster fluctuations with a low-pass filter designed to pass only those fluctuations significantly slower than a period of 65 years. To this end we apply a moving-average or boxcar filter with a 65-year window. For additional calming of much shorter periods due to the solar cycle we also apply an 11-year moving average filter.

    Each of these can be accomplished by convolving DeCl and QASI with "boxcars" of respective widths 65 and 11 years. By associativity and commutativity of convolution it suffices to convolve the two boxcars first, giving the light blue curve labeled Kernel K, a trapezoidal convolution kernel in place of the box-shaped one used for moving-average filtering. We can then smooth DeCl and QASI by convolving with K and extracting the "valid" portion, yielding respectively the green curve labeled sm(DeCl) and the magenta curve labeled sm(QASI).

    Smoothing does not preserve centering, which is a good thing because the two smoothed curves don't obscure each other yet can clearly be seen to be almost identical to within a small vertical shift (an additive constant).

    Since the resulting shape of both curves is roughly that of a 130-year cycle, twice the period of the 65-year cycle, the 65-year filter will attenuate the 130-year cycle by a factor of sinc(65/130) = 0.637. We therefore divide both curves by sinc(65/130) to offset this attenuation, thereby representing the two curves more accurately. Since sinc(11/130) = 0.988, taking that into account too would make a negligible difference so we haven't bothered to take the 11-year moving-average filter into account.

    Could the gain G be as low as 0.44 or as high as 0.48? The following graph answers this by plotting DeCL against QASI (all smoothed, centered, and divided by sinc(65/130)) for all three values of G shown in the legend.

    Clearly 0.46 gives a much better fit to QASI. This should put 1/(1-G) in the range 1.8-1.9 with good confidence.

    There has been much speculation about delays between instantaneous fluctuations in radiation and warming response. What is completely unexpected in this graph is the lack of any noticeable delay between the impact of the 130-year cycle in QASI and its counterpart in detrended HadCRUT4. Unless we are overlooking some simple explanation, this lack of visible delay seems like a great research topic.

    5. The need to include the Sun in estimating climate sensitivity.

    A common argument against CO2 as a major contributor to global warming is that climate, understood as the HadCRUT4 record of combined land and sea temperature since 1850, has not increase steadily the way CO2 does, but fluctuates seemingly at random. These fluctuations have no obvious human cause and are therefore likely to be of natural origin.

    Prior to about 1980 the expected impact of CO2 has been small compared to these presumed natural fluctuations. We have found empirically that these fluctuations can be largely removed from HadCRUT by filtering with a 65-year moving average or boxcar filter, yielding the thin blue curve in the following figure.

    The thick light green curve is 1.7*log2(CO2) indexed after filtering to average zero over the 30 years 1956-1985, that is, an anomaly. The thick light red curve is that plus the Sun, defined as 0.95*TSI*(1 - 0.3)/4. The thin dark blue curve is HadCRUT4. All three curves have been smoothed to a running mean of 65 years and then indexed to zero out their mean over 1956-1985.

    6. The hiatus in 2012 and now.

    The following graph shows monthly HadCRUT4 since 1990, in blue up to 2012 and in purple to the end of 2017. Based on the data available at the time, it forecasts future climate in respectively 2012 (thin) and now (thick) based on two trend models, respectively linear (red) and parabolic (green).
    In 2012 it was clear that a linear trend model would predict continued warming while a parabolic (i.e. quadratic) model would predict imminent cooling.

    Five years later the linear trend model's forecast had barely changed. The parabolic model on the other hand had so completely changed its forecast as to end up on the other side of the linear trend model!

    One conclusion might be that linear models predict more robustly than nonlinear. Given this, another conclusion might be that climate deniers attach little importance to robustness in choosing between models.

    So why did the hiatus happen when CO2 was rising? To answer this I detrended HadCRUT4 by the expected warming due to rising CO2 and also by the so-called Atlantic Multidecadal Oscillation. The resulting plot of climate (the red curve in the figure below) still looked pretty random.

    But then I fitted a trend line to each of the 15 decades since 1870. With the exception of the decade 1970-1980, every odd-numbered decade trended up and every even-numbered decade trended down.

    One would therefore expect that the even-numbered decade 2000-2010 would trend down. But that's in the absence of global warming. What global warming did was cancel the even-numbered down-trend.

    One would then expect that the uptrend for the odd-numbered decade 2010-2019 would add to global warming.

    Which is what all the papers have been reporting during this decade: every year hotter than the one before, and hotter than any year in recorded history!

    7. A correlation between solar cycles and 21-year climate

    The following graph plots sunspot numbers (blue) and global climate (red) for the period 1865-2000. The latter is filtered with a 21-year bandpass filter so as to reject periods significantly higher or lower than 21 years. (This rejection is not sufficient however to remove an evident 63-year cycle well correlated with the inverse of Length of Day, LOD, whose manifestation in climate has a significantly larger amplitude than this 21-year cycle, more on this later.)
    Bandpassing is accomplished by convolution of HadCRUT4 with a Ricker or "Mexican hat" convolution kernel tuned to pick out the 21-year band. Each 10-11 year sunspot cycle is numbered according to a convention established by Rudolf Wolf in 1848. Following solar max of the even-numbered cycles the heliomagnetic field (HMF) of the solar wind is polarized parallel with Earth's magnetic field, and anti-parallel after solar max of odd-numbered cycles. Earth cools during an anti-parallel HMF and warms back up after a parallel HMF.

    Another way to see the above-mentioned 21-year oscillation in HadCRUT4 is to fit trend lines to every decade or so of HadCRUT4, revealing essentially the same cycle.

    As mentioned before this can explain the so-called ``hiatus'' during the first decade of this century, which was one of these downturns, along with the subsequent very steep rise during this decade, which is currently halfway through one of the upturns.

    To date there is no agreed-on cause for this apparently strong correlation. What we do know is that Earth's magnetic field normally shields Earth from cosmic rays but this shielding is weakened when it couples to the anti-parallel HMF. One possible explanation of the 21-year climate oscillation is that water vapor molecules condense on aerosols ionized by cosmic rays, creating a nucleus for further condensation, similarly to how silver iodide seeds clouds. The resulting increasing cloud cover then cools the Earth. To date however no 21-year cycle in cloud cover has been detected. Hence either that effect is too small to be observed directly, or the explanation lies elsewhere, for example as an expression of the Gnevyshev-Ohl rule that odd-numbered cycles have more sunspots than even-numbered ones.

    The most recent downturn in climate began in 2000 at solar max of cycle 23, and ended abruptly at solar max of cycle 24 in 2012 when HadCRUT4 began a dizzying 5-year rise of 0.37 degrees. Of the various suggested explanations of the so-called climate hiatus during the first decade of the century, this one seems particularly plausible.

    This correlation can also be seen in Central England Temperature for 1659-2010, extended yet further back to 1550 by Dr. Tim Brown based on southwest winds in Devon as a proxy. Interestingly the 21-year oscillation in CET persists even during the Maunder Minimum, making it likely that the heliomagnetosphere is still active even when sunspot activity is negligible, as well as making the Gnevyshev-Ohl rule a less plausible explanation. Solar cycle numbers earlier than -5 are inferred from the oscillation in CET, whose period during the Maunder Minimum would appear to be closer to 18 years than the subsequent 21 years.

    8. Global Warming since 1850, in three stages.

    This graph plots three stages of global surface temperature (both land and sea), as recorded in HadCRUT4, a dataset assembled from many hundreds of millions of measurements made at the surface. The first stage is the 108 years from 1850 to 1958, which trended up at 0.27 degrees Celsius per century. The next stage is the half century from 1958 to 2008, which trended up at 1.26 degrees per century. The last stage is the last ten years, from 2008 to (almost) 2018, which trended up at 3.86 degrees per century.
    Detailed numbers for these plots and trends can be seen at Click on Raw Data (below the plot) for the numbers.

    9. Three natural influences on multidecadal climate

    Multidecadal climate refers to climate events that last longer than a decade. In the graph below, all faster events in the four plots have been removed with an 11-year moving-average filter, taking out the 11-year sunspot cycle, 7-year El Nino/La Nina events, coolings of at most 2-3 years due to recent volcanic aerosols, etc.
    Climate itself is taken to be HadCRUT4, the blue curve. The three natural influences plotted below it are as follows.
  • Total solar insolation TSI (red), significant only for its slow rise during 1900-1950. While small, any slow influences that one might attribute to humans must also be small during that period because there were only a third as many humans then, and moreover each human was consuming much less energy. Hence any analysis of human influences over a period starting earlier than 1950 will therefore be more accurate when it takes TSI into account.
  • A 63-year cycle often called the Atlantic Multidecadal Oscillation that a number of researchers have linked to millisecond fluctuations in the length of day LOD, whose tiny effect may be amplified considerably by magmatic volatiles as argued here.
  • The 21-year cycle pointed out in the first section (purple).
  • The human influence, CO2, is not shown. A 63-year filter removes the faster green (LOD) and purple (HMF) influences, leaving only CO2 and TSI, which as shown in the previous section accounted remarkably well for 63-year but less so when TSI was ignored.

    10-11. (These two section moved up to position -4.)

    12. The need to include TSI when forecasting with 65-year climate

    65-year climate is HadCRUT4 smoothed with a 65-year moving average filter. As the three plots in the following figure show, 65-year climate is not as well modeled by either Radiative Forcing (log(CO2)) or Absorbed Solar Insolation (ASI = TSI*0.7) alone as by both together.
    The three thick gray curves are copies of 65-year HadCRUT4 as smoothed with a 65-year moving-average filter. The three fits are as follows.
  • Radiative forcing RF (log2(CO2), blue). RF fits to within a standard deviation of 0.015 degrees (15 mK). Although this is a good fit, it is clear that RF is curving up more than climate, whence its forecast of a further rise of 2.66 degrees by 2100 is likely to be too high.
  • Absorbed solar irradiance (ASI = 0.7*TSI, red). At a standard deviation of 35 mK this is not only a bad fit but a hopeless forecaster of future climate.
  • RF and ASI together (green). This fit is accomplished using multiple linear regression. The fit improves dramatically to 5 mK, confirmed by visual inspection: the model is less curved than the pure RF model, in fact it has essentially the same curvature as the smoothed climate it is intended to model. This model forecasts a more modest rise of only 1.90 degrees, which is more believable than 2.66 degrees given the respective fits.
  • What this graph forecasts with considerable confidence is not climate in the year 2100 itself, but rather average climate, averaged over the 65 year window at 2100, i.e. the average of the 65 years 2069-2131. Climate is always changing, and the best we can say with any confidence about this graph's forecast is that about half of those 65 years will likely be hotter than forecast and the other half colder. While we can't say which of those years will be hotter, we can be pretty sure that there will be some three decades of hotter-than-forecast climate during the period 2069-2131.

    This amounts to an uncertainty principle for modern climate. If we ask for temperature over a precisely defined short period in time we should not expect much accuracy in temperature. Conversely if we ask for much accuracy in temperature we should not expect to be able to achieve it for too precise a target in time.

    13. Compound Annual Growth Rate (CAGR) of Representative Concentration Pathway 8.5

    14. 44-year trends in sea level rise for 1880-2013

    15. Amount of trash we put in the land, sea, and air, in millions of tons per year (log scale).

    In billions of tons per year, these amounts are respectively 2, 0.02, and 37. The 37 billion tons of trash we put in the air is largely the combustion products from burning fossil fuels (coal, oil, gas), not counting water vapor and aerosols (particulate matter) since those have a residence time in the atmosphere of mere days.

    Fitting CO2 and TSI to HadCRUT4

    Miscellaneous graphs, various sources, little or no annotation

    Troposphere structure as a function of latitude, and induced tradewinds.