Simple Model for rebalancing stable CO2 emissions

As requested by @richardbetts and @edhawkins here is a simple model to back up the arguments from the previous post – Stabilising Climate.

Simple Model

CO2 levels rise when the rate of change of the sources – S exceeds the rate of change of sinks – K. Without human emissions  then S = K, when averaged over one year. However with ever increasing human emissions the situation becomes dynamic

If C is the yearly value of CO2,  S  the net sources of CO2 and K the net sinks, then at time t.

C(t) = C_0 +\int_{t_0}^{t} \frac{dS}{dt} - \frac{dK}{dt}  dt

However it has been measured for at least the last 60 years that

If   \frac{d^2S}{dt^2} > 0    then   \frac{dK}{dt} \simeq 0.5 \times \frac{dS}{dt}

Now let’s assume that the world manages to stabilise annual emissions at current rates of 34 Gtons CO2/year  indefinitely.  CO2 sinks currently absorb roughly half of that figure – 17 Gtons and have been increasing proportional to the increase in partial pressure of CO2 in the atmosphere – currently that of 400ppm. Stabilising emissions now results in a decreasing fractional uptake by carbon sinks as the partial pressure imbalance between the surface and atmosphere begins to fall. The simplest assumption is that the sink increase depends only on the partial pressure difference for a given year. Therefore  if this pressure difference is reduced by half in one year then the next year it will be reduced by one quarter, then one eighth  and so on. The same argument applies for the case that it takes longer to reduce pressure difference by a half.

Year 1: 50%  Year 2: 25% Year 3: 12.5% Year 4: 6.25% etc. which is simply equal to the infinite sum

\sum_{n=1}^{\infty} {\frac{1}{2}}^{-n} = 1

So in this simplest of models, CO2 levels in the atmosphere will  taper off after just ~10 years to reach a new long term value equivalent to adding an additional one year of emissions 34 Gtons of CO2 to the atmosphere. The atmosphere currently contains 3.13 x 10^12 tons of CO2 so the net increase at equilibrium would in this simple model be just   1%. Therefore for the years following 2016 the resultant CO2 curve would look like the red curve below. If instead it takes say 4 years for the sinks to increase  by \frac{1}{2}^n then we get the blue curve. In this case it would take 30 years for CO2 levels to to stabilise and the increase would be 5 times larger.

CO2 stabilisation curves for different time constants. The red curve assumes sinks match half the imbalance in 1 year while the blue matches it in 4 years.

Posted in AGW, Climate Change, climate science, IPCC | Tagged , | 44 Comments

Stabilising Climate

In order to stop global warming all we really have to do is to stabilise CO2 emissions, not reduce them to zero!  One of the ‘myths’ promoted by IPCC climate scientists is that we have to stop burning all fossil fuels i.e. we must ‘keep it in the ground’. This is a total fallacy as I will try to explain in this post.

Carbon dioxide sources and sinks must balance once stability is reached

The origin of this belief that we must stop burning any fossil fuels by ~2050 can be traced back to Figure 10 which appeared in the AR5 ‘Summary for Policy Makers’. Here it is.

Figure 10 from SPM AR5

Figure 10 was intended to send a simple message to the world’s political leaders. Namely that there is a finite total amount of fossil fuel that mankind can safely burn, and that we have already burned half of it.  Therefore unless the major industrialised countries stop burning fossil fuels altogether by 2050, the world will warm far above 2C (red curve) causing a global disaster. This message worked, but there is so much wrong about the hidden assumptions and even subterfuge used to produce Figure 10 that I wrote a post about it at the time.

The principal assumptions hidden from view under Fig 10. are:

  1. Carbon sinks are saturating (they are not)
  2. ECS (Equilibrium Climate Sensitivity)  is 3.5C (Uncertain – and could be as low as 1.5C)
  3. Replacement of logarithmic forcing of CO2  with a linear forcing.

As a direct consequence of IPCC successful lobbying based around Figure 10, the Paris treaty now proudly “sets the world on an irreversible trajectory on which all investment, all regulation and all industrial strategy must start to align with a zero carbon global economy“. Does anyone really believe that this is even feasible, let alone realistic? It simply is not going to happen because well before then their citizens will revolt and kick them out. The best we can hope for in the short term is a stabilisation in annual global CO2 emissions.

I argue that by simply stabilising emissions, we can halt global warming. Clearly the lower total ‘stable’ emissions are then the cooler the planet will be, but even if we only managed to stabilise emissions at current values the net warming will still be <2C and CO2 levels will stop rising and stabilise at <410 ppm.

Atmospheric CO2 levels must always reach an equilibrium as the natural carbon sinks will catch up to balance emissions. For the last 40 years about half of man-made emissions have been absorbed mainly into the oceans, but also into soils and biota. The reason why CO2 levels have been continuously increasing since 1970 is that  we have been increasing emissions each year, so the sinks have never been able to catch up. Sinks will quickly balance emissions and CO2 levels will stop rising once emissions stop increasing. This fact is obvious because run-away CO2 levels have never happened in the earth’s long history. Such a balancing mechanism has always stabilised atmospheric CO2 over billions of years during intense periods of extreme volcanic activity, ocean spreading and periodic tectonic mountain building. Fossil fuels are an insignificant fraction when compared to  the buried carbon contained in sedimentary rocks.

To see how this works let’s assume that the world can stabilise annual emissions at current rates of 34 Gtons CO2/year  indefinitely. We know that CO2 sinks currently absorb half of that figure – 17 Gtons and have been increasing proportional to the increase in partial pressure of CO2 in the atmosphere – currently 400ppm. Stabilising emissions would result in the increasing fractional uptake by carbon sinks of the now fixed emissions. The remaining fraction of annual emissions that would remain in the atmosphere is therefore as follows.

Year 1: 50%  Year 2: 25% Year 3: 12.5% etc. This is simply equal to the infinite sum

\sum_{n=1}^{\infty} 2^{-n} = 1

So CO2 levels in the atmosphere will  taper off after just ~10 years to reach a new long term value equivalent to adding an additional 34 Gtons of CO2 to the atmosphere. The atmosphere currently contains 3.13 x 10^12 tons of CO2 so the net increase at equilibrium would be only an extra  1%. Therefore for the years following 2016 the resultant CO2 curve looks like this.

There is also a very good chance that we can achieve such a fixed limit, rather than pretend to meet an impossible target of zero emissions. However this  does mean that CO2 levels will remain at 404 ppm indefinitely, which is far higher than a planet without human beings, but that still leaves us plenty of time to replace fossil fuels with new nuclear energy. Furthermore such a strategy would save trillions of dollars from being wasted on the pipe dream of renewable energy.

Controlling CO2 levels by stabilising emissions also has one other advantage. It means that we will eventually be able to control the level of ‘enhanced global warming’, thereby avoiding another devastating ice age which otherwise is due to begin within the next 5000 years.


Posted in AGW, Climate Change, climate science, IPCC, Science | Tagged , | 49 Comments

Is the Supermoon responsible for record low Polar ice formation?

The  “Supermoon”  on November 14th coincided with the closest (perigean) approach to the earth of the moon since 1948. Tidal forces are inversely proportional to the cube of distance. Full moon occurs  when the sun lines up with the moon and November 14th is also close to the perigee of the earth’s orbit round the sun. This combined to produce strong tides. At full moon in November the moon lies in the southern hemisphere . So for the Arctic it is the opposite facing tides that is strongest while for the Antarctic it is the direct lunar facing tide that dominates. They are symmetric. Here are my calculations of the tractional acceleration at different latitude covering November, based on the JPL ephemeris.


The tractional tidal force acting at 45N (green) and 75N (blue). Polar regions experience far greater tidal ranges during the lunar month than temperate regions. The Supermoon amplified this effect by about 20%. This enhanced tidal mixing and has probably inhibited ice formation since early October.

It is the tidal range that is maximised during a perigean spring tide. That is the difference between low and high tides. At high latitudes this effect is magnified  as just one tide dominates and neap tides effectively disappear completely. This gives an extreme varying monthly tidal range. Tides act throughout the ocean,  dragging both deep  and shallow water alike. This increasing churning tidal flow since October has had two effects. First it has  inhibited natural sea ice formation, and secondly mixed in more warmer water from lower latitudes than normal.  These large tidal ranges look likely to continue till the end of 2016 before returning to normal.


The Arctic ocean also has relatively shallow basins with narrow channels at the Bering Sea and to the North Atlantic between Iceland-Scandinavia. This accentuates tidal flow.

Thanks to @Kata_basis for prompting me to look into this !

see also:

  1. Does the moon trigger interglacials
  2. Evidence of  tidal effect on the polar jet stream
  3. Tidal effects in Polar regions
  4. The straw that broke the camels back



Posted in Climate Change, Ice Ages | Tagged | 5 Comments