Global Warming for Electrical Engineers

An electrical analogy to the flux of energy from the surface of the Sun as it reaches and then leaves the Earth’s surface on its journey into deep space.

I haven’t written much about global warming lately, but I have noticed that the resurgence of the ‘alt-right‘ seems to have emboldened people to express ‘sceptical’ views.

People expressing these views are in general no more or less stupid than anyone else. However, they do fail to understand that their own competence in one area, or the popularity of their views in polls, has no bearing on the correctness or otherwise of their understanding of anthropogenic global warming.

In a recent interaction with a Nameless American, it became clear that despite being able to assemble the facts, this individual was unable to understand the basic process by which the surface temperature of the Earth comes to be what it is. And hence they could not understand why it is rational to expect that increased amounts of carbon dioxide in the atmosphere are affecting the surface temperature of the Earth.

So here, for that Nameless American, is Global Warming for Electrical Engineers: Apologies to everyone else.

Basic Circuit

Figure 1: A simple electrical circuit. The key feature is that the same current flows through both resistors R1 and R2.

The basic circuit required to understand the way in which the surface temperature of the Earth is established is shown in Figure 1. Two key features of this resistor-divider circuit are that:

1. The current flow through circuit elements R1 and R2 is the same.
2. The DC steady state operating point of the circuit is determined just by the resistances and the voltage of the DC power supply

Now the analogy we will make is this:

• Voltage is analogous to temperature: In the same way that voltage differences drive electrical currents, temperature differences drive energy flows.
• V0 is like the surface temperature of the Sun
• V1 is like the surface temperature of the Earth
• V2 is like the temperature of the deep space – almost absolute zero.

Importantly, the only way to get thermal energy on or off the Earth is by electromagnetic radiation – mainly visible and infrared light.

Notice that the surface temperature of the Earth is determined (in the steady state) by the requirement that the average flux of energy onto the Earth’s surface is the same as the average flux off the Earth’s surface.

This is analogous to the way Kirchoff’s current law is used to establish the steady state DC voltage V1.

Figure 2: We are drawing an analogy between the flow of electrical current through resistors in series and the flow of energy from the Sun onto the Earth’s surface and then secondly off the Earth’s surface and out into space.

How the analogy works

The surface of the Sun is hotter than the Earth: Radiation travels from the surface of the Sun through space and arrives at the top of the atmosphere.

For the moment let’s forget about the radiation reflected from the cloud tops, and consider only radiation which travels through the atmosphere and reaches the Earth’s surface. We’ll discuss the effect of this assumption in Subtlety #2 below.

The radiation which travels through the atmosphere is mostly visible light – the sunlight which warms the Earth’s surface.

The resistance R1 then determines the amount of heat delivered to the Earth’s surface from the Sun’s Surface. The actual value of R1 is determined by factors such as the distance from the Sun to the Earth.

Now we consider the re-radiation of thermal energy from the Earth’s surface. This is in the form of infrared light. In the same way that warming happens mainly on the ‘day’ side of the Earth, cooling happens mainly on the ‘night’ side of the Earth.

Whereas the atmosphere is mainly transparent to incoming radiation, the atmosphere is mainly opaque to infrared radiation. If we humans could see at the relevant the wavelengths, and looked up at the night sky, we would not see the stars, but just a ‘fog’.

The atmosphere would appear to be totally opaque at these wavelengths: but it is not.

When we shine light into a fog, it is multiply scattered and only a tiny amount of light makes it out the other side of the fog. We can consider the fog as presenting an impedance R2 to the transmission of radiation out to the heat sink of deep space.

The surface temperature of the Earth is determined in the steady state by the requirement that it is hot enough to ‘drive’ infrared radiation through the impedance R2 and out into space.

• If the Earth’s surface temperature is ‘too low’, more energy will arrive on the surface of the Earth than leaves and the surface temperature will rise.
• Similarly, if the Earth’s surface temperature is ‘too high’, more energy will leave the surface of the Earth than arrives and the surface temperature will fall.

Eventually, a steady-state is reached: a dynamic equilibrium. The temperature of the Earth’s surface becomes hot enough, that it glows brightly enough to drive sufficient infrared radiation through the Earth’s atmosphere and out into space.

In our circuit, this is equivalent to the voltage V1 rising until it reaches a value sufficient to drive the operating current  through the resistor R2.

Anthropogenic global warming is caused by an increase in the impedance R2. For a fixed surface temperature, this reduces the amount of radiation which leaves the Earth’s surface and reaches space.

In order to re-establish the equilibrium and drive the requisite energy flux through the atmosphere and out into space, the temperature of Earth’s surface needs to rise

Subtlety #1

The electrical analogy in Figure 1 is ridiculously simple, so let’s make it more complicated and (very slightly) more realistic.

Figure 2: Ra, Rb etc represent transmission through the atmosphere in different wavelength bands.

Figure 2 replaces a single resistance R2 with an array of parallel resistances each of which radiatively couples the surface of the Earth to the coolness of space. We could imagine that each parallel resistance represents (say) transmission in a different wavelength band.

The important observation is that increasing the impedance any of Ra, Rb etc always increases, the total impedance R2. Since the current through the circuit is fixed, this will cause an increase in the voltage V2.

Considering our Earth analogy, if we decrease the transparency of the atmosphere to infrared light in any waveband, this will increase the overall impedance. Since the flux of radiation onto the Earth is unaffected, this will cause an increase in the surface temperature of the Earth.

Of particular interest is the radiation which leaves the Earth’s surface in wavelength bands that are absorbed by carbon dioxide molecules.

Aside on Subtlety #1: ‘blocked bands’

The conclusion of the previous section is that if we make the atmosphere more opaque in any wavelength band, the surface temperature of the Earth will increase. This conclusion is inescapable. Unless…

…The only time that increasing a number makes no difference to the number’s value is if that number is already infinite.

So global warming sceptics frequently argue that ‘the carbon dioxide bands are blocked‘. They argue that carbon dioxide absorbs infrared light so effectively, that at certain specific wavelengths the atmosphere is (practically) 100% opaque.

Their argument is that increasing the amount of carbon dioxide cannot therefore increase the opacity of the Earth’s atmosphere any further. The problem with this argument is that it is ‘just wrong‘.

[We can see why in various ways, but firstly I feel compelled to note that water vapour is dramatically more effective than carbon dioxide at blocking infrared light, and yet sceptics don’t apply the same argument to water vapour!]

The actual mechanism of transmission of infrared light through the atmosphere is complex: it is illustrated schematically in Figure 3.

At infrared wavelengths the atmosphere looks ‘foggy’. Radiation travels through ‘fog’ in a process involving multiple scatterings – think of car headlights shining into fog: some of the light comes back in the direction towards your headlights and some goes forward and sideways.

Figure 3: Illustration of the energy flux onto and off the Earth’s surface. On average, roughly 240 W/m^2 of solar energy reaches the Earth;s surface. This is re-radiated as infrared red light at wavelengths at which the atmosphere is opaque. The light is scattered, and some comes back to the Earth, and some makes its way further up the atmosphere. Eventually the light reaches a height – typically 6 km to 10 km – where it can radiate freely into space.

This process of multiple scattering goes on repeatedly until radiation makes it to an elevation of typically 6 km to 10 km above the Earth’s surface at which point the atmosphere is thin enough to allow radiation directly out into space.

Importantly, there are no ‘completely blocked bands’. If there were, our satellites would fly over the Earth at night and find no emission at all at some wavelengths: that is not what is seen.

What is seen is the ‘top of the fog’: the radiation from the highest part of the ‘fog’. Radiation at all wavelengths does eventually make its way through the atmosphere in a process of multiple random scatterings.

Increasing the concentration of carbon dioxide makes the atmosphere less transparent in some wavelength bands and, as we saw in the previous section, that inevitably drives an increase in the temperature of the Earth’s surface.

Calculating the sensitivity of the Earth’s surface temperature to an increase in carbon dioxide concentration is complex, but in fact our estimates have not changed much since Arrhenius’s first estimate in 1896.

Arrhenius calculated that doubling carbon dioxide concentration from the historical value of 280 ppm  to 560 ppm would cause an increase of 4 °C.

Now using supercomputers and complex climate models we estimate this sensitivity to be 3 °C ± 1.5 °C. The robustness of this estimate in the face of the overwhelming additional calculational complexity is testament to the fundamental simplicity of the physics involved.

Subtlety #2

We can model this by splitting R1 into two series resistances describing transmission from the surface of the Sun to the top of atmosphere (R1a) and subsequent transmission through the atmosphere to the surface of the Earth (R1b). The equivalent circuit diagram is shown in Figure 4.

Figure 4. Modification of the equivalent circuit to describe reflection from the top of the atmosphere.

In this modification, R3 describes the reflection of light from the top of the atmosphere.

• If there are lots of white cloud tops during the day, then R3 is small: it is small compared with the sum of R1b and R2, But notice that clouds at night don’t affect R3.
• If very little light is reflected from white cloud tops during the day, then R3 is large compared with the sum of R1b and R2.

In practice, on average the flux of energy from the Sun is 340 watts per square metre at the top of the atmosphere, and about 100 watts per square metre are reflected into space. This indicates that R3 is approximately twice as large as the series sum of R1b and the components of R2.

Clouds

The reason I mention this additional complexity is because of the role of clouds. It is important to look at clouds from both sides, from up and down and from night and day.

The inevitable warming caused by increasing carbon dioxide concentrations will inevitably cause changes in the amount of water vapour in the atmosphere. And these changes can affect the pattern of clouds formed on Earth and give rise to effects which alter R1b – the transmission of visible light between the top of the atmosphere and the surface of the Earth.

Roughly speaking, additional cloudiness during the day could cool the Earth, reducing the warming effect. But additional cloudiness at night will warm the Earth.

On balance the effect is difficult to calculate, but our best estimates result in warming consistent with that observed experimentally.

Conclusions

This article is written for one individual: the Nameless American  who thinks that his cleverness and popularity means that their’gut belief’ that global warming is a hoax is correct. They are, sadly, ‘just wrong’.

I too would love to believe that global warming is a hoax, but it isn’t.

The electrical models I have described could be improved by adding some capacitances to the circuit to allow the dynamics of the changes in temperature to be simulated.

These electrical capacitances, would be analogous to the heat capacity of the top layers of the land and ocean surfaces of the Earth.

But there is not much point: scientists have done this calculation and the results are in. We have already made the measurements, and the results are in also.

The real argument for the ‘alt-right’ is this: if you think the economic benefits of burning unlimited coal and emitting unlimited carbon dioxide outweigh the costs: please make this argument. I disagree with you, but it’s a fair argument.

But don’t attack the science. Our understanding of this process is a collective triumph for humanity.

[November 21st 2016: Weight this morning 73.5 kg: Anxiety: High: off the scale]

2 Responses to “Global Warming for Electrical Engineers”

1. Ed Davies Says:

Shouldn’t R1 really be a constant-current source? Changing the temperature of the Earth’s surface won’t appreciably affect the amount of short-wave radiation received from the Sun, only the amount of heat sent in the opposite direction.

The big simplification here, I think, is ignoring convection which basically thermally bypasses the IR opacity of the atmosphere. The effect of GHGs is to raise the effective radiation surface (“photosphere”) of the Earth so that the adiabatic lapse rate resulting from expansion and compression in the convection cycles has more distance to work over so raising the surface temperature.

2. protonsforbreakfast Says:

Mmmm. Thank you for bringing your insight to my odd presentation.

On your second point, ‘yes’, but in my defence I was seeking simplification.

On your first point, ‘I don’t think so’. IF you follow the analogy, the surface of the sun is at ~ 5500 K and the Earth’s surface is at ~ 290 K – so the resistance ratio R1/R2 is – crudely speaking – around 20. In other words changing R2 dramatically won’t change R1+R2 very much which makes the circuit effectively a current source.