Battery Day: First Results

March 20, 2021

Me and my new Tesla. This unit contains 13.5 kWh of battery storage along with a climate control system to optimise battery life. We have placed it in the porch so that (when visits are allowed again) everyone who visits will know about it!

Last September, Tesla held their ‘Battery Day‘ during which they unveiled their road map towards cheaper, better, batteries.

Not to be outdone, last Monday VW held their own ‘Battery Day‘ during which they unveiled their road map towards cheaper, better, batteries.

And last Thursday was my own battery day, when Stuart and Jozsef from The Little Green Energy Company came and installed a Tesla Powerwall 2 at Podesta Towers in Teddington. I was (and still am) ridiculously excited.

I am still evaluating it – obviously – but here are a couple of notes.

How it works

The system has two components. An intelligent ‘gateway’ that monitors loads and supplies, and a climate-controlled battery storage unit.

Click for a larger version. The left-hand graphic shows how AC power enters our house, and how DC power generated by solar panels is linked to the grid. When the Solar PV is sufficient ,power is exported to the grid. The right-hand graphic shows how the TESLA ‘gateway’ device monitors the solar PV, domestic loads and battery status and intelligently decides what to do.

The gateway (and the battery) are electrically situated between the electricity meter and all the loads and power sources in the house. So all energy enters or leaves the battery module as AC (alternating current) power.

But its internal batteries must be supplied with DC (direct current).

This makes it ideal for storing power from the AC grid, but less than ideal for storing the DC current generated by solar PV panels.

One might have expected that a device designed to store solar power might intrinsically operate using DC and indeed, some battery systems – positioned between the solar PV and the inverter – do this.

So the choice to place the Powerwall™ where it is, is a compromise between the extra functionality this location offers – it can back up the entire house – and the inefficiency of storing solar PV which is first converted to AC by the inverter, and then re-converted back to DC by the Powerwall. The support document states that the conversion from AC to DC and back to AC has 90% round-trip efficiency.

The photograph below shows the gateway installed under the stairs in our house.

Click for a larger version. The Tesla ‘Gateway’ installed in our house. The unit is positioned in between the electricity meter and all the domestic loads. The black conduit leads under the floor to the battery which is installed in the porch.


The system is controlled by an app which is – frankly – mesmerising. It shows how electrical power flows between:

  • the grid,
  • the battery,
  • our home, and
  • our solar panels

Click for a larger version. Screenshots from the app at various times yesterday.

There is less room to adjust the parameters of the system than I had anticipated. This appears to be because, in exchange for a guarantee that the battery will retain at least 80% capacity (10.8 kWh) after 10 years, one is required to relinquish detailed control to the Tesla Brain.

Through a built in network connection, the device is in constant touch with Tesla who monitor its performance and can detect if it is abused in some way. I am not sure how I feel about that – but then guaranteed long-term performance is certainly worth something.

One feature of this relinquishing of detailed control concerns ‘time-of-use’ tariffs. I anticipate that – especially in winter – I will need to charge the battery overnight on cheap rate electricity.

The system supports this mode of operation but is not yet operational. Apparently it needs to study the patterns of household use for 48 hours before being enabled.

When operational, one gives the system general instructions and then allows it to choose when, and by how much, to charge. There is for example no way to force the battery to charge to 100% on command.

In practice I suspect it will be fine, but at the moment it still feels a little weird.

Performance on Day#1

The simplest way to show how the Powerwall™ works is by looking at the data which the ‘App’ makes available.

The first graph shows the household demand through the day. It’s fascinating to look at this data which has 5 minute and 0.1 kW resolution. The metrologist in me would like more – but in honesty, this is enough to understand what is happening.

Click for a larger graph. See text for details.

Now we can look to see how that demand was met. Overnight, we relied mainly on the grid.

Click for a larger graph. See text for details.

The battery could have supplied this overnight electricity, but it had been set to hold a reserve of 16% of its capacity (~2 kWh) in case we required backup after a power cut. We have lowered that setting now because, thankfully, power cuts are rare in Teddington. The battery drew power from the grid overnight in two short periods to maintain this reserve.

Additionally, at the end of a sunny day in which the solar PV filled the battery, there was brief period where we returned electricity to the grid.

During the day – which was very sunny 🙂 – the household electricity demand was met by the electricity from the solar panels.

Click for a larger graph. See text for details.

Without the battery, most of this 16.91 kWh of electricity would have been sent to the grid. But now only a tiny fraction was returned to the grid, most of it being captured by the battery – see below.

Click for a larger graph. See text for details.

The graph above shows the battery maintaining its reserve charge at night, and then charging from the solar PV during the day. At peaks of household demand, the charging is paused. At around 16:00, the battery was briefly full, and shortly thereafter it began discharging to meet household demand.

As I write this at 1:00 p.m. on the day after the day shown (a rather dull day 😦 ), the battery is 56% full and charging.

The graph below shows all the above curves together.

Click for a larger graph. See text for details.


The Powerwall system is an object of wonder. It is beautifully engineered and miraculous in its simplicity.

It transforms the utility of the solar PV allowing me (rather than electricity companies) to benefit from the investments I have made.

I will post more about the performance in terms of cost, electricity and carbon dioxide when I have more data.

But for the moment I will just thank Jozsef and Stuart from The Little Green Energy Company for their professionalism and attention to detail. And ‘No’. I am not being paid to say that – quite the opposite!

Stuart and Jozsef from The Little Green Energy Company. You can’t see it, but they assure me they were both smiling. Click for a larger version


Sometimes I find it hard to like EDF

March 16, 2021

Click for larger version. EDF wrote to me today to say they are increasing the price of ‘night rate’ electricity by 73.7%.

Energy is a wonderful thing. But sometimes it can be hard to like the companies which sell it to us…

…especially when they increase the price of their product by 73% overnight!

As regular readers will know, since 2018 I have been working hard to reduce my household energy consumption and the concomitant carbon dioxide emissions.

My 3-step plan has been:

  1. Reduce household heating requirement with insulation and triple-glazing.
  2. Switch from gas heating to electrical heating with a heat pump.
  3. Use solar panels and a battery to generate low-emission electricity and reduce the cost of switching to electrical heating.

Part#1 is complete: winter is not yet at an end, but heating demand appears to about 50% lower.

Part#2 is underway and I hope to have a heat pump installed this summer. But electrical heating is more expensive than gas heating.

Part#3 is underway: the solar panels are performing well and a battery should be installed this Thursday. The battery should allow me to use mainly my own solar electricity, or EDF off-peak electricity for most of the year.

I carried out extensive modelling of the effect of varying patterns of electricity consumption and compared different ‘tariffs’.

I had based my costings on the fact that the night rate for electricity would be about 5p/kWh and day rate would be about 25p/kWh. Of course I knew these costs could vary over time.

Nonetheless, it would be an underestimate to say that I was ‘disappointed’ when EDF wrote to me this morning to say that price of night time electricity was to rise from 4.99 p/kWh to 8.67 p/kWh…

…a 73% rise!

Like I said, sometimes it can be hard to like the companies which sell us energy.


I am thinking about it.

But switching is, in my opinion, a distraction. It is a way of distracting us ‘fish’ from the fact that we are in a ‘barrel’ and at the mercy of the confusopolists.


Previous articles about the house.




The Cat Sat on the Mat

March 14, 2021

While walking though Teddington the other day I saw tender sight which brought a smile to my eyes.

A man was mending his very old car – an Austin Maxi – and had tools and components laid out around the car.

And just by the car was a small mat on which his cat was very contentedly sat.

I commented to him that it was very considerate of him to put down a mat for the cat.

He smiled.

Then he told me that the mat was there to cover a drain so that he didn’t accidentally lose any parts. And the cat was sitting there opportunistically rather than by invitation.


Correlation does not imply causation

It had seemed so obvious that the man had placed the mat down for the cat. I had immediately intuited his state of mind and fondness for his cat.

In order to have fully appreciated what was happening, I would have needed to:

  • Imagine into being a drain – for which I had no evidence – it was completely covered.
  • And then understand that it would be sensible to cover the drain if working near it – a mat would be ideal.
  • And finally understand that cats will sit on mats unbidden.

And so I was reminded that even the simplest and most apparently obvious things are sometimes not what they seem.


See also these links suggested by astute commenter Dave Burton

COVID-19: What next?

March 14, 2021

Friends, if you are anything like me, you are most probably sick to death of this virus. And yet it goes on.

I have been looking back at my posts over 2020 and I still feel OK about most of my comments. But as we re-open schools I thought it would be an idea to look back at where we were last September 2020 when schools re-opened.

By 26th September 2020 it was obvious that the exponential growth in prevalence which marked the start of the second wave was underway. At the time I did not imagine that twice as many people would die in that second wave as in the first. I wrote: Here we go again.

I mention this because looking up the prevalence data today (below) I notice that the viral prevalence is currently greater than it was when schools last re-opened.

Click for a larger version. This

We are probably all more versed in anti-viral protocols than we were in September 2020, but nonetheless we have to expect some increase in viral transmission and prevalence.

The difference between then and now, is that now 35% of the UK population have had a single-shot of vaccine and a substantial number of the most vulnerable are fully immunised.

We should thus expect that rising cases will not necessarily lead to the same rise in hospital admissions and deaths that we saw previously.

What’s happening now?

The ONS prevalence data (above) necessarily lags what’s happening on the ground by about two weeks.

If we look at current cases, we can see that there has been a rise in cases – or at least a slowdown in the rate of decline –  in recent days.

An unknown fraction of this rise is definitely an artefact of the large number (more than 1 million per day) of lateral flow tests being done around the return to schooling. (See the first 20 minutes of this Independent Sage Presentation for an explanation). But the exact fraction is unclear.

Click for a larger version.

So at this point it is hard to know what this rise means: whether it could just be another wiggle. Or whether it could be – and I am being serious – the start of Wave 3.

If we look at the long term graph of positive cases, we see that back in July 2020 (~Day 191) when prevalence and cases were 10 times lower, cases suddenly began to rise and kept rising.

Click for a larger version.

Hospital Admissions and Deaths did not start to rise for another 8 weeks.

Click for a larger version.

This time – the vaccine makes things different – and in 8 weeks time a full 50% of the entire population should have had at least a first shot. So we should not expect a rises in cases to lead to rises in hospital admissions and deaths.

But if the viral prevalence becomes widespread again then we risk breeding variants that can spread under the new conditions in which they find themselves.

For completeness, here are the data on hospitalisations and deaths

Click for a larger version. The pink shading shows that hospitalisations have now fallen below their trend since late January 2021.

Click for a larger version. The pink shading shows that deaths have now fallen below their trend since late January 2021. The blue line shows 1st-shot vaccinations and the light blue line shows 2nd shot vaccinations . Dotted lines show projections and trends.

COVID-19 Milestones

March 4, 2021

Friends, allow me to highlight three upcoming COVID milestones.


  • First the positive: Tomorrow I will receive my first shot of vaccine. Hurray!
  • Second, and more tragically, tomorrow or Saturday will mark the date on which the death toll from the second wave will reach 82,732 which is double the death toll from the first wave. Words fail me.
  • Thirdly, Monday will mark the re-opening of schools. And two weeks later we will be able to see the effect on the rates of COVID cases.

Intriguingly, the populations who will be mixing (directly or indirectly) after schools re-open – parents, teachers, and children – will be mainly unvaccinated. But the most vulnerable people will be mainly vaccinated.

Measuring the resultant extra cases and the way they feed through to hospital admissions and deaths will likely be critical to the development of the Government’s ‘re-opening’ strategy.

Update on Cases

As I am sure you know the data are looking good – the daily rate of positive tests for COVID-19 is falling with a halving-time which may be even less than 16 days.

Click for a larger version.

The number of positive cases per day is shown above on a linear scale. The red dotted curve shows the rate at which cases fell after the first lockdown.

One can see that just over two weeks ago (~day 410) the rate at which the cases were falling reduced, but the rate of decrease now appears to be even faster than before – but we only have a few days data.

It is easier to see the trends if the data is plotted on a graph with a logarithmic vertical axis. On this graph (below) exponential trends show up as straight lines.

Extrapolating the trend of the last few days, we might hope for around 1000 cases per day by Easter (4th April). At that level, track, trace, support and isolate really should – after a year of utter failure – be able to cope.

However, we should more reasonably expect the rate at which cases are falling to slow (to some extent) after the schools re-open.

Click for a larger version.

Update on Hospital Admissions

The rate at which people are admitted to hospital is also falling, but unfortunately the data file of the ‘dashboard’ website is corrupted.

As you can see in the screenshot below for today 4th March 2021, the daily figure is 757. But in the data file, this figure is recorded as being appropriate to the 28th February. So using the data file would lead one to think that cases were falling more rapidly than they are.

Click for a larger version.

Using daily figures as they were reported, I find that hospital admissions are falling with a halving time of 21 days – surprisingly the same as after the first lockdown.

However the recent daily admissions data looks to have ‘steps’ in it – and given the errors above – it may not be reliable.

Click for a larger version.

Update on Deaths 

The rate at which people are dying from COVID is falling, faster and faster.

The figure below shows that after falling with an initial halving-time of about 16-days (orange dotted curve), daily deaths are now falling even faster.

The difference between the extrapolated 16-day halving time trend and the actual data is shaded in pink.

This is almost certainly the vaccine taking effect.

Click for a larger version.

Also shown in blue is percentage of the entire UK population (adults and children) who have received a first dose of vaccine.

The rate appears to have slowed a little in the last week or so – presumably as second-doses are prioritised.

Nonetheless, the rate is impressive: roughly 50% of the population will have received a first shot of vaccine by Easter.


So here we are… in the sad shadow of a gigantic second wave of deaths.

But with a little luck, the army of elderly vaccine gladiators which I will join tomorrow will vanquish the Covidian contagion before it can mutate.

Death to the Covidians!

COVID-19: The Johnson Plan

February 22, 2021

Friends, I just thought I would note down my first reflections on the latest plan of our Glorious Leader.

I actually skipped his TV announcement tonight to instead watch the NASA press conference releasing the video of the landing of Perseverance on Mars.

And the mindset of the engineers who made that landing happen is, by chance, relevant to our situation.

They achieved success by considering all the ways in which their mission could fail, and made a plan that avoided each failure mode in turn.

In other words, they achieved success by a relentless focus on failure.

This mindset does not seem to be part of the Government’s strategy.

Why do I say that?

Because announcing in advance that their policy will be “irreversible” is a mistake in the same category as calling The Titanic “unsinkable”.

The Johnson Plan

My first thought is that this would be nice if it all worked, but I find it hard to believe that it will – it seems too optimistic.

With regard to my comments above on the mentality that enabled NASA’s successful Mars landing, there appears to be no plans for failure. No plans with the structure of:

  • We expect X
  • If X doesn’t happen we will do Y.

Let me explain why I think that.

Yesterday I said the Government’s self-proclaimed focus on ‘data not dates’ should focus on monitoring the rate of daily positive cases. This is because hospital admissions and deaths follow cases.

So setting criteria for lifting restrictions based on cases enables a rapid response before hospital admissions and deaths follow.

Click for larger version. See text for details. Daily rate of positive Covid-19 tests since June 2020. The vertical axis is logarithmic so exponential rises and falls look like straight lines. The dotted lines show my extrapolations of the trend of the data.

The graph above shows how the daily case rate has varied since June 2020.

I have highlighted various events in amber ovals to show how clearly this graph speaks to us.

  • The first oval marks the ‘opening’ of society: pubs, restaurants and hairdressers. 
    • There were 600 cases/day
    • Cases immediately began to rise marking what is clearly the start of the Second Wave which has killed about 80,000 people (twice as many as the First Wave), brought untold misery to many, and devastated the economy more than many wars.
  • The second oval marks the return to school. 
    • Cases immediately rose from 1400/day to 2600/day in a week and kept rising.
  • The third oval marks the return to Universities. 
    • Cases immediately rose from 6200/day to 17500/day in a week and kept rising.

With the exception of Lockdowns, every Government action to tackle this was ineffective

So much for the past, what about the future?

The Johnson plan

Step 1 of the J-Plan calls for a widespread Return to School on 8th March 2021.

  • As the fourth oval shows, cases will then optimistically be in the range 4000/day to 9000/day, much higher than when children last returned to school in September 2020.
    • I find it hard to believe that this will not stimulate an immediate rise in cases, exactly as it did last time.
    • Test, Trace, Support and Isolate is still not functioning. Indeed it could never reasonably have coped with that many cases anyway.
    • If cases rise, will the Government reverse its irreversible course?
  • The fifth oval shows, the earliest date (12th April 2021) at which regular retail, personal care and outdoor hospitality will re-open.
    • This seems to predicated on case rates in the range 1000/day to 3000/day.
    • This rate is much higher than the case rate of 600/day when retail last opened in July 2020
  • The sixth oval shows the earliest date (17th May 2021) for Step 3. .
    • There is no point thinking about this because it’s just so unlikely.

The Impact of Vaccines

Of course the difference between ‘now’ (Ovals 4 & 5) and ‘then’ (Ovals 1, 2 and 3) is the role of vaccines.

If vaccines break the chains of viral transmission, then the Government’s plan may make sense, but I think that is unlikely to happen until later in the summer when the younger adult population and children are vaccinated.

More likely is that (in the next few months) the vaccines will reduce the burden of illness arising from cases. So hospital admissions and deaths may continue to fall even as cases rise.

This leaves open the possibility that for several months, viral prevalence may remain very high, leaving open the door for viral mutations with unknown properties. They could be benign, or deadly.

If the NASA team were planning this, they would see this as a deadly failure mode and plan for it.

My fear is that the Government will just hope for the best, but not make a plan to cope with it. Announcing in advance that your plan is “irreversible” does not inspire me.

COVID-19: Next Steps

February 21, 2021

Friends: This is how epidemics work:

The VIRUS causes CASES which cause HOSPITAL ADMISSIONS which cause DEATH

Click for a larger version. Remember that on a logarithmic graph, a straight line means that a quantity is increasing or decreasing exponentially.

The graph above shows the number of positive tests for COVID-19 per day, the number of hospital admissions per day, and the number of COVID-19 deaths per day since April 2020.

The link between these three quantities is clear.

Currently everything is terrible.

  • There are hundreds of people dying every day
  • Hospitals are full and many people with other illnesses cannot be treated.
  • The economy is badly damaged with consequent distress and unhappiness.
  • We can’t even hug the people we love.

But things are getting better.

  • The vaccines are here.
  • All the pandemical indicators are falling.

On the eve of the Government’s announcement of its ‘Get out of Lockdown Strategy’ in this post I will summarise my hopes and fears.


The graph below shows the number of positive tests for COVID-19 per day since July 2020.

Click for a larger version. Remember that on a logarithmic graph, a straight line means that a quantity is increasing or decreasing exponentially.

We see that currently the number of positive cases per day is halving roughly every 16 days – this decline is significantly faster than after the first wave.

  • Since the virus is now apparently more transmissible than the First Wave variant, this is a surprising fact.

However even if the trend continues, we will not be at the level at which First Wave restrictions were lifted (about 600 cases per day in July 2020) until May 2021.

With hindsight we can see that cases began to rise IMMEDIATELY Lockdown#1 ended, and this was indeed the start of a very large and deadly Second Wave.

So I hope the Prime Minister will:

  • Set a level of cases (say 1000 cases a day) that can trigger the lifting of particular sets of restrictions.
  • Make the test, trace, and isolate system work, and employ targeted isolation and local ‘ring’ vaccination campaigns around cases.
  • Set a second level of cases (say 2000 cases a day) that can trigger the re-imposing of particular sets of restrictions.
    • It is imperative that we do not let the virus run away again, and the first sign of viral spread is rising numbers of cases.

What I fear is that the Prime Minister will:

  • Set a level of ‘trigger’ cases (say 5,000 cases a day) which is too high for test, trace and isolate to cope with
  • Do nothing to rectify the flaws with test, trace and isolate.
  • Not announce in advance what will happen if cases rise.
  • Use the vaccination programme – which should have a detectable effect on deaths soon – to be used to as an excuse to allow lifting of restrictions.

However allowing the virus to spread amongst younger less-vulnerable people is a recipe for breeding strains of the virus with new properties.

Recall that currently nobody on Earth knows why COVID-19 affects older people more. It is perfectly conceivable that a new strain might cause harm to younger people. We should guard against any such possibility.


The graph below shows the number of hospital admission for COVID-19 per day since July 2020.

Click for a larger version. Remember that on a logarithmic graph, a straight line means that a quantity is increasing or decreasing exponentially.

Once again it is obvious in retrospect that once admissions stopped falling at the start of August – about a month after cases began to rise – the Second Wave was with us.

Nothing the Government did (with the exception of National Lockdowns) made any difference. Remember that roughly 25% of those admissions resulted in death.

This graph makes it clear that it is the virus which is still in charge.

So I hope the Prime Minister will:

  • Announce that after restrictions are lifted (based on a cases criterion), any rise at all in hospital admissions will cause a re-imposition of restrictions.

This is essential so that hospitals do not once again become COVID-hospitals.

One last curiosity is that  hospital admissions are falling at the same rate – roughly halving every 21 days – as after the First Wave.

  • Since the cases are falling faster, this indicates an increasing ratio of admissions to cases – I do not know why.


The graph below shows the number of COVID-19 death per day during the Second Wave.

Click for larger version

Like cases, but unlike admissions, deaths are falling with a halving-time of roughly 16 days – rather faster than after the First Wave.

  • I do not know why!

I don’t have the skill to make a model to simulate how the protective effect of the vaccine will translate into lower death rates. But qualitatively I would expect the death rate to plummet shortly.

My hope is that the Prime Minister will:

  • Ignore the death rate in setting criteria for re-opening.

My reasoning here is simple: until we have full population vaccination, wherever the virus goes, cases, hospital admissions and then deaths will follow.

So increasing cases and admissions are omens of impending increases in the death rate.

  • The counterargument to this is that if death rates fall – because of vaccination – then the harm caused by the virus is eliminated, so why not lift restrictions?
  • The reason is that if COVID-19 is widespread in the population it will likely mutate under selection pressure. Only if we drive the prevalence of the virus to extremely low levels can we minimise the risk of a harmful mutation arising.


Currently everything is terrible, but things are getting better.

My recommendations are (I hope) clear, but I do not live with the same pressure that the Prime Minister has to.

And with regards to schools in particular, the pressure to re-open is immense.

From a pandemical point of view, I would say it is too early to re-open schools: the prevalence is too high and re-opening will almost certainly increase the spread of the virus, which test trace and isolate will probably fail to contain.

And re-opening risks the breeding of a new Summer Term strain of virus with potentially appalling properties.

However the Prime Minister may consider this a chance worth taking to re-start education.

I’m glad I don’t have that job.


Guy Callendar: Precis of his foundational paper on Global Warming

February 21, 2021

Click for a larger version.

Earlier this week the inestimable Ed Hawkins reminded the world that February 16th was the 83rd anniversary of a breathtaking publication in 1938 by Guy Callendar in the Quarterly Journal of the Royal Meteorological Society.

You can read the 18-page paper here: The Artificial Production of Carbon Dioxide and Its Influence on Temperature

A year or two ago I wrote extensive notes on the paper intending to write about it on this blog, but somehow in the chaos of my retirement those notes have disappeared and I no longer have the fortitude to attempt that again.

But it is such a singular paper that it is worth pointing out its general structure and one or two of its highlights.

But before we get onto the paper, it is worth mentioning Guy Callendar’s job title: Steam Technologist to the British Electrical and Allied Industries Research Association . In short his foundational paper on what we now call global warming involved expertise way outside his ‘day job’.

The Abstract

The abstract is short and clear.

  • He points out that in the 50 years from 1887 to 1937 humanity had put 150,000 million tonnes (150 gigatons (GT)) of carbon dioxide into the atmosphere and that most of it was probably still there.
  • He then calculates the effect of this extra carbon dioxide in terms of the extra “sky radiation” that falls on to the Earth.
    • I don’t know if his use of the term ‘sky radiation’ was its first use, but clearly it was not standard terminology.
    • He estimated the entire surface of the Earth was warming by 0.003 °C per year, or 0.15 °C over the previous 50 years.
  • He then points out that meteorological measurements are consistent with his claim.

In short, he understood the entirety of the mechanism underlying global warming, and had the insight to realise how it might be detectable.

And a recent review of his paper using modern techniques has confirmed its accuracy.

The Start

Click for larger version

I particularly like the beginning where he states the basic problem.

The scale of the Earth’s natural weather systems are so large and the amount of energy involved so enormous that it is difficult to conceive that anything that any human intervention could possibly have any effect.

Today this still remains astonishing. The energy that humanity uses collectively each day – all the electricity and gas and oil burning – is (roughly) only 1 part in 10,000 of the natural daily flux of energy on and off the planet.

So the assertion that changes in the atmospheric concentration of an unreactive gas present only in trace quantities could possibly have any effect  seems similarly astounding.

But here he states it directly up front: this is what he intends to show.

The Structure of the Paper

He makes his argument in 6 sections

  1. The Rate of Accumulation of Atmospheric Carbon Dioxide
  2. Infra-Red Absorption by Carbon Dioxide and Water Vapour
  3. Sky Radiation
  4. The Effect of Carbon Dioxide on Sky Radiation
  5. The Relation between Sky Radiation and Temperature
  6. The Observed Temperature Variations of the Earth

I do not have energy to follow through the details of his arguments, parts of which are subtle and complex.

So instead I will highlight just a few features in each of these sections which delight me.

1. The Rate of Accumulation of Atmospheric Carbon Dioxide

In the early part of the 20th Century, measuring the atmospheric concentration of carbon dioxide was a major challenge.

But Callendar takes what few measurements there were from 1905 and 1930-36 and notes that the atmospheric concentration of carbon dioxide appears to have increased.

He then compares the measured increase with what he would have expected to observe based on known emissions from burning coal and oil (which would increase the atmospheric concentration), and the expected rate at which carbon dioxide would dissolve in the oceans (which would decrease it).

He gets good agreement between the measurements and his calculations but modestly observes:

“Such close agreement with the calculated increase is, of course, partly accidental.”

Comments such as this endear him to me.

2. Infra-Red Absorption by Carbon Dioxide and Water Vapour

This is a very technical section covering transmission of infrared light through the atmosphere. Measurements in those days were crude by modern standards.

But he correctly identifies the critical roles played by both water vapour and carbon dioxide in absorbing infrared light of different wavelengths.

The end result is a table showing the fractions of energy emitted by the Earth in different infrared wavebands for three different temperatures of the Earth.

When attempted in full, this calculation is enormously complex. A few years ago I spent five (!) blog articles describing the calculation!

  1. Light Transmission through the atmosphere
  2. Light Transmission through a gas
  3. Light Transmission through the atmosphere
  4. Feedback and Climate Models
  5. What was that all about

Sadly, Callendar did not have access to MODTRAN

3. Sky Radiation

If we stand on the Earth then on a clear night we can look out to space using our eyes, and the transparency of the atmosphere to visible light allows us to see the stars.

But it is different for infrared light.

The key point of Section 2 was that at certain infrared wavelengths, the atmosphere is opaque. If we could see in these wavelengths the atmosphere would look like a mist.

Also, in the infrared, the Earth is not dark but glows with a brightness that depends upon how hot it is.

Because  the atmosphere is partially opaque in the infrared, some of the infrared glow from the Earth is absorbed by the atmosphere and re-radiated back to the Earth’s surface, warming it. This is what Callendar calls “Sky Radiation”.

“Sky radiation” is analogous to the way visible light is scattered back to its source in a mist.

  • In a very dense mist shining headlights into a fog can dazzle the driver because lots of light is returned within a short distance in the direction it started out in.
  • In a less dense mist, the light can travel quite a distance and the reflection back to the driver is not so bright.

Callender calculates the fraction of the infrared light radiated by the Earth that is returned as ‘sky radiation’.

He considers many factors that affect the amount of sky radiation received back on Earth.

  • Altitude: mountains
  • Latitude: he considers the poles, temperate and tropical locations
  • Angle: he considers the effect of radiation from near the horizon and radiation from directly above, the zenith.
  • The composition of the atmosphere – the nearly constant concentration of carbon dioxide, and the highly variable concentration of water vapour.

He then significantly underplays the difficulty of the calculation.

Click for larger version.

4. The Effect of Carbon Dioxide on Sky Radiation

This is another very technical section, but the gist of it is simple.

Having calculated previously (Section 3) how much ‘sky radiation’ was being returned to the Earth, in this section he calculates how this would be affected by changes in the concentration of carbon dioxide.

He has several key insights.

First he considers only infrared light with wavelengths between 0.013 mm and 0.016 mm. This is because in this waveband the Earth’s infrared glow is at its brightest, and it also corresponds to a range where carbon dioxide absorbs strongly. So it is in this waveband that changes in the atmospheric concentration of carbon dioxide will have their greatest effect.

Secondly, he notes that the atmosphere is already totally opaque to infrared light in this waveband. So the effect of increasing the absorption further (by adding more carbon dioxide) is to lower the effective height above the Earth from which the ‘Sky radiation’ is returned to the Earth. Lower parts of the atmosphere are warmer and so re-radiate more ‘sky radiation’.

He writes:

Click for larger version.

His calculation is summarised in his Table 5 which I have annotated below

Click for larger version. Annotated version of Callendar’s Table 5. The small net increase in ‘sky radiation’ on doubling the atmospheric concentration of carbon dioxide arises from an increase from the lowest atmospheric layers and decrease from the upper atmospheric layers

His calculations indicate that doubling the concentration of carbon dioxide in the atmosphere will give rise to a 0.58% net increase in the amount of sky radiation the waveband between 0.013 mm and 0.016 mm.

5. The Relation between Sky Radiation and Temperature

Now we come to the key question:

What amount of warming might we expect from this apparently tiny 0.58% increase in the amount of sky radiation in the waveband between 0.013 mm and 0.016 mm caused by doubling the amount of carbon dioxide in the atmosphere?

Callendar states the key insight clearly at the start.

Click for a larger version.

The warming of the Earth by the Sun is apparent and obvious to all. The more subtle question is where does that heat go?

Ultimately, if the temperature of the Earth is constant, then ALL the solar radiation which warms the Earth must be re-radiated out into space. As Callendar puts it:

…because no other type of heat exchange is possible.

After some subtle arguments about the role of water vapour, Callendar summarises his calculation with a graph which I have annotated below.

Click for larger version.  I have drawn in red the current concentration of atmospheric carbon dioxide (417 ppm as I write in February 2021) and so the predicted temperature rise since 1938, is roughly 0.8 °C.

The graph shows the expected change in the mean surface temperature of the Earth as the concentration of atmospheric carbon dioxide changes.

The curve allows Callendar both to look ahead to suggest the magnitude of future warming, and to look back to see whether warming has taken place in the previous 50 years

Looking back is the focus of the final section of the paper, but in this  section he looks ahead to future centuries.

In Table 6 he summarises his expectations based on annual carbon dioxide emissions of 4.3 gigatonnes.

Click for larger version

I have sketched his expectations from Table 6 (above) on the figure below.

In fact carbon dioxide emissions are currently growing 8 times faster than he anticipated, and at this rate a century of Callendar’s anticipated changes takes only 12.5 years.

Click for larger version. The graph shows atmospheric concentration of carbon dioxide in parts per million versus year. The green circles show Callendar’s estimate of future concentrations based on 4.3 gigatonnes of emissions per year. Actual emissions are currently around 35 gigatonnes of carbon dioxide. The black curve shows modern best estimates of historical carbon dioxide concentrations.

6. The Observed Temperature Variations of the Earth

While clever calculations and insightful fancies about infrared radiation in the atmosphere are one thing, and bold predictions about the temperature of the entire Earth are another – actually determining whether the average temperature of the Earth’s surface had warmed by a few tenths of a degree Celsius is quite another.

Callendar set about the task by consulting a tome published by the US Smithsonian Institute : World Weather Records

The problems with the strategy are obvious:

  • There is a great deal of natural variability in weather records with time scales of days, weeks, months, years (obviously) and decades!
  • Not all records are reliable.
  • Very few records are continuous across the period of interest and few are of a long length.
  • Some locations on Earth have very little data and so warming or cooling may be either under or over represented.
  • The local environment around many stations has changed over decades due to urban growth which might give rise to spurious warming

But Callendar understood that there were strategies for getting a reliable result in the face of all these difficulties.

Most critically he understood that because each station samples its local environment differently from each other station, that one should look at only changes (so-called anomalies) of each station record with respect to itself.

In other words he asked how the 10 year average from weather station 1 had changed over decades, and how the 10 year average from weather station 2 had changed over decades – he didn’t try to average the raw data from the weather stations. Using anomalies eliminates many possible systematic errors.

He also had a big advantage. The climate change he was looking was global! Thus he expected a similar trend from ALL tropical stations, ALL temperate stations, and ALL polar stations. This allowed him to compare trends deduced from different combinations of stations to see if the trends he saw were really present everywhere, or just statistical artefacts.

After extensive testing and tabulation he came up with the graph below.

Click for larger version. Estimates of changes (anomalies) in the temperature of the Northern and Southern temperate zones, the tropics, and of the whole Earth. These estimates are based on analysis of data from 147 weather stations.

Like all good scientists, Callendar includes a table of data with his final result. This has enabled me to overplot his estimate (in green) with modern re-analysis of many more stations by the Berkeley Earth team.

Click for larger version. The Berkeley Earth estimate of the Global Average temperature from 1850 to 2018. Overplotted in green is Callendar’s estimate of the changes of Global Average temperature from 1880 to 1935. I have offset Callendar’s data by -0.4 °C to take account of the use of different baselines.

Actually – the result is pretty good. The failure to detect the exceptionally warm years of 1876 and 1877 was probably due to poor coverage by weather stations.


What impresses me most about his paper is the completeness of his understanding from the basics, through the multiple competing processes at play in the atmosphere, to the likely climatological consequences.

At that time, papers were ‘read’ aloud (I think) and members of the Royal Meteorological Society asked questions which were minuted and included in the paper

Callendar was clearly aware of many of the detailed criticisms raised by his interlocuters, but he does not get lost in those details. His response begins beautifully:

Click for a larger version.

Here he points out that the details don’t matter. ANYTHING that makes the atmosphere more opaque to infrared radiation MUST warm the Earth no matter how complex the mechanism.

In response to a request for details of the natural movements he responds that he has written a paper on this but it was 8 times longer than the present paper!

Click for a larger version

So his astonishing insight and endeavour was communicated with modesty in the face of what I read to be quite a polite but unwelcoming reception by meteorologists who likely thought “What does this Steam Man know about meteorology!

The challenge with any precis is to actually use less words than the original document. With that in mind I must stop writing now. My hope is that this article may help you make your own way through the paper. I will let Guy Callendar’s conclusion be my own.

Click for larger version.

COVID-19: The risk to men

February 17, 2021

Click for a larger version. Output from the QCOVID calculator showing the risk arising from ethnicity and biological sex compared to a 61-year old woman.

Friends, you may have read recently (e.g. The Guardian) of a new tools for analysing the risks of dying from COVID-19 including many different risk factors, including ethnicity.

The tool – called QCOVID – is available for you to use here.

Media coverage generally highlighted the relative risks appropriate to different ethnic minorities.

For some reason, the media did not mention the most prevalent risk factor for dying from COVID-19: being male.


As one does, I immediately typed in my own basic statistics: I am a white male aged 61 with a BMI of 24.8. The result was this:

Click for a larger version. Output from the QCOVID calculator for someone with my vital statistics.

I took note of the absolute risk of a COVID associated death: my risk was 0.0173% over a 90-day period i.e. a risk 1 in 5,780.

I then changed the ethnicity and biological sexuality entries to the calculator across a number of categories.

Changing ‘my’ biological sexuality to female I saw that ‘my’ risk went down to 0.0077% over a 90-day period i.e. a risk 1 in 12,987.

So the additional risk factor associated with being a biological male was 2.2.

Click for a larger version. Output from the QCOVID calculator showing the risk arising from ethnicity and biological sex compared to a 61-year old woman.

I was curious as to how this compared with differences arising from ethnicity and the results relative to the ‘Me-female’ are shown in the table above.

I found the results striking.

  • For females, the additional ethnic risk factors ranged from 1.2 to 2.0. In comparison the additional risk factor of being a white male was 2.2
  • In every ethnic category, being male carried an additional risk factor varying from 2.2 to 4.6 compared to the equivalent female.


I did not investigate all the various categories in the QCOVID calculator, so I cannot confirm the complete generality of this result.

But from my simple investigation, I conclude that:

  • Men of any ethnicity are at least twice as likely to die from COVID-19 as women from the same ethnic background.
  • The additional risk faced by women associated with their ethnicity is less than the additional risk faced by white males associated with their sexuality.

My question is this: why has this dramatic result affecting millions of people not been reported more widely?

Update 19th February

A correspondent pointed out in the comments this excellent data source

This site shows that excess mortality amongst men occurs worldwide, but the extent of it is highly variable from one country to another.

I don’t have an explanation of this phenomenon, but it does appear to be very real, whatever its cause.



Covid-19: Quick Update

February 15, 2021

Click for a larger version. The number of positive CV cases per day shown a daily data (dots) and as a 7-day retrospective average. The data re plotted on a logarithmic axis. Presented in this way, exponential trends appear as straight lines. The rates of decline are highlighted and extrapolated by dotted lines. It is clear that the rate of decline of cases has changed – it is now halving every 16 days  – much faster than the initial rate of halving every 21 days, and faster than it did after Lockdown#1.

As I have mentioned before (1, 2) the daily rate of positive cases is falling faster than it did after previous lockdowns.

As you can see on the graph above, the halving-time for cases is now 16 days rather than roughly 21 days after the first lockdown.

The difference has been sustained for 4 weeks which is long enough to arise from something other than chance.

I cannot tell if it arises from the vaccination programme, but if the trend continues…

  • …we will reach 1000 cases per day in mid-April rather than mid-May

This is significant. 1000 cases per day is probably the maximum rate that our embarrassing Test, Trace and Isolate system might be expected to cope with.

So – good news! But I wish I knew why!

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