What is the most impactful way to use a square meter of sun-exposed surface area to mitigate climate change?

The Question

What is the most impactful way to use a square meter of sun-exposed surface area to mitigate climate change, measured over the next 20 years?

Imagine you have a spare square meter of sky-facing surface that you are dedicating to serving our collective climate future (you’re SO generous).  How can you use that m² of solar flux to most efficiently reduce the temperature of our planet? 

The Assumptions

Let’s start by assuming you have full agency over the use of this m², that the (substantially) only source of energy input is the sun, and that you’re only allowed to have 2 major conversions of potential energy (e.g., you can’t make electricity from a photovoltaic and then use it to run a compact fusion device that you then use to turn CO₂ and water into sugar that you put in a still to displace your morning habit of drinking vodka that is flown in daily from Poland).  Let’s also assume that you live in an ‘urban’ environment.  Finally, let’s assume that the time frame of relevance is 20 years (if we don’t make some serious progress by then… then we’re probably having a different conversation anyway. Also, it’s unreasonable to expect the lifetime of any of our surface schemes to exceed 20 years without needing repair). 

The Metric

To compare our various options, we need to define a figure of merit.  For this exercise, let’s pick a new unit called Integrated Radiative Forcing Equivalent (IRFE).  What we care about is the amount of integrated global warming (over 20 years, see The Assumptions) that will be reduced per m² of area.  You can use kilocalories (kcal) per m² as the units (since a calorie is, definitionally, the energy required to raise 1g of water by 1 degree Celsius), and you can easily convert to kWh m^-2 with the conversion factor of 0.00116222 if you’re more used to reasoning with that unit.  We’ll use kWh m^-2 for the purposes of this exploration.

As an example, imagine that we have a solution (Solution X) that decreases the net energy flux on earth by 1 Watt per m² of surface area.  Over 20 years (or ~6.307*10⁸ seconds), this will amount to 6.307*10⁸ Joules or ~150,741 kcal per square meter.  Therefore, our Solution X has an IRFE of 150,741 kcal m^-2 or 175.32 kWh m^-2.  We can also identify an Average Radiative Forcing Equivalent (ARFE) as the average power per m² over the 20 year period.  In the above example, 1 W m^-2 is the ARFE.

An additional unit will be useful to us, which is to define the ratio of average net energy flux reduced to the average energy flux of sunlight on that square meter. Let’s call this unit the Radiative Forcing Gain (RFG), a unitless number.  Let us consider the ‘standard square meter average flux’ a measurement of the amount of sunlight, averaged over a year, that hits the earth.  Let’s use the value of 250 W m^-2 to keep it nice and round (see Solar Flux).  Therefore Solution X has a RFG of 0.004 (or 1 W m^-2 / 250 W m^-2).  See Defining Units for an overview of this example.

In practice, our posed question can then be restated to: how can we maximize ARFE, IRFE, and RFG?

One last metric will come in handy: the global temperature change associated with the identified solution.  To translate between radiative forcing and temperature change, for small perturbations we can used the simplified formula of

$$\Delta T = \lambda \Delta F$$

where $\Delta F$ is the change in radiative forcing (in W m^-2), $\Delta T$ is the change in average surface temperature (in K), and $\lambda$ is the climate sensitivity parameter.  We will use a value of 0.45 K W^-1 m² for $\lambda$.  $\Delta F$ is a measurement of average radiative forcing across the surface of the earth, so in order to calculate a $\Delta T$ from an ARFE, we will need to divide the ARFE by the surface area of the earth.  Put another way, for Solution X, $\Delta F$ would be the change in radiative forcing achieved if every square meter of Earth was covered in Solution X.  See Constants and Assumptions. 

Therefore, the global change in temperature associated with a constant ARFE of 1 W m^-2 is about -9 * 10^-16 K.  If we were to cover the entire surface of the planet with this solution, we would achieve a temperature reduction of ~0.45 K.  

The Contenders

So… what can you do with that sun-exposed m² that you’ve so generously donated to our collective cause?  Let’s list some options:

• Do nothing
• Paint it white
• Install an optimal emitter
• Install a photovoltaic cell
• Install a solar heater
• Plant grass
• Grow a tree
• Plant food 
• Install a photocatalytic oxidation device
• Install a Direct Air Capture device

Do Nothing

Let’s start with ‘Do Nothing’.  It’s the baseline case against which we need to measure.  It’s also the easiest thing to do (doing something is 100% harder than doing nothing).  It’s tricky to associate an ARFE with this passive approach, so forgive the abductive reasoning.  Urban land use is estimated at 1.5M km² of a total of, approximately, 149M km² of land and 510M km² of total surface area of the planet.  Of that, approximately 0.182M km² of roofs and 0.121M km² of roads exist (as of 2005).  Roofs have an average albedo of ~0.12.  Total global warming over a 20 year period associated with the Urban Heat Island effect is estimated to be between 0.06 and 0.11K.  Let’s use the lower end of that estimate, or 0.06K, and equally distribute it across all urban land use (1.5M km²).  Inverting our radiative forcing calculation (see Do Nothing, which features a bastardization of the Radiative Forcing model as described in Defining Units) gives an ARFE of -44.9 W m^-2 and an RFG of -0.18 (meaning it contributes to net warming).  This does not take into account any changes from the initial state of the system before the introduction of this m² of available land (e.g. if you deforested land to build your house).^*It’s hopefully obvious to the reader, but this calculation is done on an average basis – and your own property is almost certainly not average.  This caveat holds across the analysis of this document.

Paint it White

The simplest thing we might think to do is to paint the surface with a reflective coating to increase its albedo.  For example, if we were to paint our roof white, the net change (on average) of albedo would be from ~0.12 to ~0.65.  One might expect that this would mean that we increase the amount of radiation back out of the atmosphere by (0.65 – 0.12)*250 W m^-2 for an ARFE of 132.5 W m^-2 and an RFG of 0.53.  Unfortunately, as is so frequently the case with issues of climate, it’s not that simple.  Not all of that reflected radiation goes back into space.  Some of it is absorbed in the atmosphere, and, if this kind of intervention was deployed on a global scale, the net impact on the temperature as a function of elevation in the atmosphere could change cloud formation patterns and more.  It has been debated back and forth, but for the sake of this document we’ll use Jacobson’s model which reveals a net effect of a 0.07K temperature rise of painting the Earth’s roofs white.  Again inverting our radiative forcing calculation (see Paint it White), this means that painting a roof white has a net ARFE of ~-432 W m^-2 and an RFG of -1.73.  Translating this to a unit which we can carry forward based on albedo change, the ARFE per unit of albedo change is ~-814 W m^-2. This means that, in urban environments, increasing albedo is net harmful.  We’ll keep this impact separate in future scenario contemplation since there is still much debate on this topic.  

Note that our ‘Paint it White’ analysis did not take into account additional effects like reducing the cooling loads of buildings in summers and increasing heating loads in winters.  

Install an Optimal Re-emitter

Ok, so the complexity of the global climate system made our ‘Paint it White’ plan questionable.  What about if we built an optimal re-emitter?  In the extreme case, for the sake of a clean hypothetical framework, imagine that we could convert the incoming solar spectrum to an optimal ‘re-emitted’ spectrum that could perfectly penetrate to space without interaction or interference with the atmosphere / clouds / etc.?  While clearly non-physical, we can easily evaluate this case to set the bounds of possibility.  Let’s start with the upper-limit on ARFE and RFG.  We can first imagine that all incoming radiation is re-emitted back to space at optimal wavelengths with 100% efficiency of conversion between wavelengths.  Thus, it is possible to emit 250 W m^-2 of radiation back into space.  One could also conceive of an idealized emitter that also enables the radiation of earth’s finite temperature (call it 298K) back into space (2.7K) through an equally idealized window of wavelengths that is in perfect communication with space.  According to the Stefan-Boltzmann law, the radiation per unit surface area per unit time is proportionate to T⁴.

$$\frac{P}{A} = \sigma T^4$$

$$\sigma = 5.67 * 10^{-8} W m^{-2} K^{-4}$$

Therefore, in equilibrium with space, a m² of surface at the temperature of earth will net emit^*Note that this number exceeds the average solar flux.  If it weren’t for the greenhouse effect that we’re so concerned about, our planet would be cooling itself off at a dramatic rate.  It’s imbalance of fluxes that is the issue. ~447 W m^-2.  Under this absurdly idealized scenario (that I acknowledge as impossible), the extreme limit to total IRFE is then 250 + 447 or 697 W m^-2 for an RFG of ~2.8.  More realistic analysis (which provides for finite transmissivity of the atmosphere) shows a limit of closer to ~100 W m^-2 net cooling from an ideal surface of tailored emissivity for a total ARFE of ~350 W m^-2 and an RFG of ~1.4.  Again, we don’t take into account the impact of the net cooling on, for example, an annual heating bill, nor have we taken into account the impact of albedo change compared to Doing Nothing on the Urban Heat Island effect.  See Optimal Emitter for the calculations.

Install a Photovoltaic Cell

Given the hubbub about renewable energy and climate, the most obvious choice for what to do with a spare m² might be to install a solar panel.  The impact on net solar flux will be twofold: first, a direct albedo effect as described above in the ‘Paint it White’ section, and, second, the potential reduction in GHG emissions associated with the use of renewable electricity.  To evaluate this scenario, we will need to make several assumptions.  Let’s start by presuming that the lifecycle emissions of a solar cell are zero (meaning that it took no emissions to manufacture the panel – clearly wrong) leading to an emissions intensity of generated solar electricity of 0 kg CO₂ kWh^-1.  We need to also assume an emissions intensity of the energy that we are displacing with electricity generated in a solar panel.  Let’s assume the average emissions intensity of the US grid, or about 0.45 kg CO₂ kWh^-1.  We also need to assume an efficiency for our solar cell.  Let’s start with 20% – a reasonable single junction efficiency for 2020.  Finally, let’s assume that we can perfectly store and use the electricity such that 100% of what is generated goes to displacing electricity that would otherwise be sourced from the grid at the average grid emissions intensity.  We can translate mitigated-emissions into a reduced concentration of CO₂ in the atmosphere by taking a (presumed linear) average of the total emissions reduced in the 20 year period and calculating a change in concentration (all else being equal) of CO₂ in the atmosphere.  From this, we can leverage an approximate relationship between the atmospheric concentration of CO₂ and the radiative forcing due to this gas with the equation

$$\Delta F = 5.35 * ln \frac{C}{C_0}$$

where the units of $\Delta F$ are, of course, W m^-2.

However, for every 1% of CO₂ we avoid putting in the atmosphere, the concentration of atmospheric CO₂ avoided will be less than that 1%.  The partition factor of CO₂ into the atmosphere (i.e. the fraction of emitted CO₂ that ends up in the atmosphere) is an important quantity.  Over long time spans, the number approximates 0.45 (for a ton of CO₂ emitted today).  However, over 20 years, we will approximate the partition factor at 0.60.

See PV Cell for the calculations (including the assumptions around the total change in concentration of CO₂ in the atmosphere per kg of emissions which translates to the radiative forcing for CO₂).  An ARFE of ~996 W m^-2 and an RFG of ~4.0 are calculated, but feel free to play around with the assumptions therein.  If your local community uses coal for electricity and you install high-efficiency solar panels, much higher ARFE’s are possible.  

PV panels have an albedo of ~0.05.  Thus, according to our calculated ARFE per unit of albedo change (from an average albedo of 0.12) of ~-814 W m^-2 we would expect a 814*(.12-.05) or ~57 W m^-2 additional benefit to ARFE of installing solar panels (under presumption that they will be installed in an area subject to the Urban Heat Island effect) for a total of ~1053 W m^-2.

In the PV Cell document two other means of calculating ARFE have been employed as a sanity check to the original calculation.  One employs Absolute Global Warming Potential (AGWP) of CO₂ and the other leverages the concept of Radiative Efficiency.  For thorough discussions of both, see the IPCC AR5 Chapter 8 Supplementary Materials.  

Install a Solar Heater

A solar heater is a device that (typically) absorbs solar radiation as heat and then uses this heat for the purposes of space heating or water heating.  Therefore, the analysis for PV Cells can be easily modified to accommodate space heating.  The key differences will be the efficiency of conversion (let’s assume a 70% efficiency) of sunlight into heat and the emissions intensity of the energy they are displacing (let’s assume natural gas at ~0.18 kg CO₂ kWh^-1).  These assumptions translate to an ARFE of ~1394 W m^-2 and an RFG of ~5.6 (without albedo).  With albedo, ARFE is brought up to ~1452 W m^-2.

Why are these values higher than for PV?  The efficiency of PV is low compared to the efficiency of modern heat engines for grid electricity production.  Installing higher-efficiency solar panels, or displacing higher emissions-intensity electricity generation, would lead to higher PV ARFE.

Plant Grass

Now we get into some less obvious options for how to use our m².  What about planting grass?  We are leveraging nature’s ability to convert solar energy and CO₂ into biomass through photosynthesis.  To perform this analysis, we’ll need to make assumptions about:

• The fraction of light that encounters the canopy
• The photosynthetic efficiency
• The fraction of the biomass that is permanently sequestering carbon

Starting with the most generous set of assumptions of 100% light capture, 100% permanent sequestration of photosynthate, and a photosynthetic efficiency reflecting the best of what grasses can do of 3.7%, we can calculate the amount of CO₂ a m² of grass can pull out of the atmosphere per unit time. 

See Plant Grass for the calculations.  Without taking albedo change into account, an ARFE of ~135 W m^-2 and an RFG of 0.54 are calculated.

Why are these numbers so much lower than for a PV cell or a solar heater?  Well, for starters, photosynthesis just isn’t that efficient.  Further, heat engines are less than 100% efficient, meaning that for every kWh of fossil electricity we reduce, we reduce more than a kWh of fossil energy consumption (which is proportional to GHG generation).  

But our above assumptions were… generous.  Taking a different approach to calculating ARFE, we can look at average yields of fast-growing grasses (like miscanthus) to calculate how much CO₂ is really being captured per m² per unit time.  Based on empirical data, CO₂ capture rates are yet still lower than calculated from our ideal case above by a factor of ~7.  This top-down calculation of ARFE yields ~21 W m^-2.  However, this is a pessimistic view since not all photosynthate results in harvested biomass (i.e. some goes to root production or to root exudate that ends up in the soil).  Therefore, the real number is probably somewhere in between these cases.  See Plant Grass for these calculations.  

Plant a Tree

Instead of planting grass, why not plant a tree?  The US Forest Service and the FAO have both calculated average carbon uptake for a variety of forest types as well as for afforestation (the process of planting a forest where there wasn’t one previously).  The ranges of carbon sequestration are widely varying by forest type, but a reasonable value to assume is 2.75 kg m^-2 year^-1.  From this number we can replicate our Plant Grass analysis (see Plant a Tree) to calculate an ARFE of ~14 W m^-2 and an RFG of 0.06.

This is yet lower still than our grass calculation!  We can make sense of this based on the efficiency of grass crops vs. trees.  However, there is a critical assumption for our grass crop that we don’t have to make for the tree: what fraction of the biomass stays sequestered.  You have to harvest the grass and make sure it doesn’t find its way back to CO₂ to secure the ARFE benefit.  With a tree, no worries!  Just make sure it doesn’t burn down.  Or get eaten by bugs.  Or chopped down and left to rot. Or…

Plant Food

Even better, let’s plant something that we actually want to consume!  The analysis will follow the logic of the Plant Grass section, but with some additional assumptions embedded.  We’ll need to assume the fraction of the plant that goes into edible biomass, the emissions intensity of the calories that we are displacing in our diet, and emissions associated with any additional inputs to the plant (e.g. fertilizer).  Let’s start by assuming that 100% of the plant biomass is edible (one could imagine having grown algae) and that the average calorie being displaced is the US average calorie. Let’s also assume that any residual biomass is ‘recycled’ back to CO₂ (meaning that it isn’t sequestered), and that any inputs required have a zero-emissions footprint.  

Under this set of assumptions, and further generously assuming high photosynthetic efficiency (3%) and high light capture efficiency (100%), we could hope to produce ~6.5 kcal per m² per hour, or ~56,500 kcal per m² per year.  At 2600 kcal per day for the average American, that’s almost 22 days of food or 6% of your annual food consumption.

About ¼ of the average American’s GHG footprint comes from food and agriculture.  There are many contributions to this assessed value, including the emissions intensity of inputs (i.e. fertilizer), the emissions associated with the supply chain, methane from ruminants, etc.  To simplify, on average about 2.6 kg of CO₂e are emitted for every 1000 kcal consumed in America.

Using the calculated calorie displacement rates of our m², we can calculate mitigated CO₂e emissions and translate that to a calculated ARFE (see Plant Food).  At an ARFE of ~751 W m^-2 and an RFG of ~3, planting food is starting to look like a great option.  But those were some pretty optimistic assumptions we made. 

Coming at it from another angle, the most productive food crops (on a calories per acre per year basis) include the sweet potato, sugar beet, and sugar cane at a rough global average productivity of 7 kcal per m² per day of edible biomass.  This translates to an ARFE of ~34 W m^-2 and an RFG of ~0.14.  Even if we look at relatively high productivity rates achieved in certain regions, we get a factor of ~3 improvement… a sobering sanity check on our above calculation for the ideal case.  

Why is there such a wide gap between ideal and realized?  First, our assumption of photosynthetic efficiency was… generous.  For real food crops, the average photosynthetic efficiency is closer to 1-2%.  Additionally, for our food crop calculation we chose crops that have substantial inedible biomass generated (meaning that energy from the sun is being used to make parts of the plant that we don’t ultimately eat).  Finally, we didn’t take into account any environmental or pest pressure on the crops that results in reduced yield.  

Install a Photocatalytic Oxidation Device

So far we’ve looked at solutions to directly emit energy back into space (thus lowering the heating burden on our atmosphere), on solutions that displace our use of resources that otherwise are associated with GHG emissions, and on solutions that actively remove CO₂ from the atmosphere.  But CO₂ isn’t the only GHG that matters.  Methane and nitrous oxide are two potent GHGs (at 86x and 268x higher GWPs than CO₂ over 20 years).  What if we could figure out a way to use the energy from the sun to remove methane or nitrous oxide from the atmosphere?

Methane is a GHG that has a finite lifetime in the atmosphere (~12 years on average) because it is thermodynamically favored for methane to oxidize to CO₂.  The majority of this oxidation occurs in the atmosphere due to the reaction of methane with hydroxyl radicals that were generated by high-energy photons interacting with the atmosphere.  

We will define a photocatalytic device as a device that leverages incident photons to catalytically oxidize methane to CO₂ or to catalytically react nitrous oxide to a low-GWP form of nitrogen.  One such example of a device is a photocatalytic oxidation device that leverages solar energy to generate hydroxyl radicals that can convert methane into the less potent CO₂, effectively accelerating the natural process of methane oxidation in the atmosphere. 

There are many assumptions that we need to make to analyze such a device.  To start, we need to assume what fraction of incident solar radiation we capture (100%), what fraction of that solar radiation is suitable for photocatalysis (in this case, we’ll assume that it’s only the UV radiation, or ~5%), and the efficiency of the photocatalytic reaction that generates a hydroxyl radical (7%).  We further need to assume what fraction of the generated hydroxyls will go towards oxidizing methane, as opposed to reacting with some other species in the atmosphere (15% – the percentage of naturally-generated hydroxyls in the atmosphere that do just that).

Under these sets of assumptions, about 0.2 kg of methane are oxidized per m² per year.

Because methane behaves differently in the atmosphere than CO₂, there is a different model for the relationship between concentration and radiative forcing.  We’ve implemented this model in Photocatalytic Oxidation. The 0.2kg of oxidized methane per m² per year, over a 20 year period, is equivalent to an ARFE of ~132 W m^-2 and an RFG of ~0.5.  The document also includes two other methods to calculate the ARFE, leveraging the radiative efficiency of methane and the GWP of methane.  

This ARFE is larger than that of ‘Plant Grass’ or ‘Plant a Tree’, but is lower than that of some of our other solutions.  Why, if methane is so much more potent a GHG than CO₂ over a 20 year time frame, does this solution yield a lower ARFE than, say, a solar panel?  By our assumptions above, the total solar energy to hydroxyl radical oxidation efficiency is ~0.05%.  Compare this to the solar energy to electricity efficiency of a solar panel of ~20%.  This factor of 400 more than makes up for the ~86x improvement in GWP reduction of tackling methane instead of CO₂.  As a heuristic argument, if the average grid electricity generation device is 60% efficient (on a carbon basis), then, roughly, 667x more CO₂ will be mitigated in a solar panel than methane will be mitigated by the photocatalytic oxidation device herein described.  If we divide by the GWP effect, this translates to a ~8x expected ARFE from a PV cell than from a photocatalytic oxidation device.  At a calculated ~1053 W m^-2 ARFE of our PV cell, this is exactly what we find!

Is there anything we could do to get back that order of magnitude?  We could 1) increase the fraction of the solar spectrum that can participate photocatalytically, 2) increase the quantum efficiency of the photocatalyst, or 3) increase the fraction of hydroxyls that go towards oxidizing methane.  If we were to bring up the total efficiency to, say, 0.5% by increasing the fraction of the solar spectrum that can participate from 5% to 50%, then our ARFE would be ~1315 W m^-2.

Install a Direct Air Capture Device

With all of the complexities of the solutions listed above, we might consider ourselves at a loss to determine by what limit our analysis might be constrained. Thankfully, the techno-optimists have delivered (or at least promise to deliver…) us a solution that is conceptually simple: pulling CO₂ out of the air directly with a Direct Air Capture (DAC) device. DAC takes CO₂ directly out of the atmosphere and concentrates it for sequestration. The thermodynamic limit for the efficiency of capturing and concentrating CO₂ from ambient (~400ppm) conditions is ~.13 kWh kg^-1. The analysis shown in DAC assumes implemented efficiencies that map onto the best contemporary assumptions of where DAC might get (~$100 per ton of captured CO₂). We will presently ignore the question of what we might do with all of the concentrated CO₂ once we have it.

The Summary

So… of the considered schemes for using a spare m² of area in an urban environment, what’s the most impactful way to decrease integrated radiative forcing?

By the above analysis, the rank ordering follows (sorted by the nominal scenario – see the worksheet for the high and low scenario assumptions):

Of course, what we might care about is $/ARFE if we have abundant space but finite capital. In lieu of a thorough analysis, you can find assumed costs for implemented solutions in the Summary tab of the worksheet. Feel free to make a copy of the document to put in your own cost assumptions. My finger-in-the-air assumptions based on my own experience in the field are populated and the results are shared below for the nominal scenario.

Perhaps unsurprisingly, the less capital-intensive solutions fare better in an ARFE per dollar assessment. But not all solutions are created equally – some solutions (like planting food, installing a PV panel, or installing a solar heater) provide value that would otherwise have been paid for. Consequently, there is an assessed offsetting ‘negative cost’ associated with the value provided by these solutions (note, in this very back-of-the-envelope attempt to bring $ into it – no discount rate, financing, or any other reasonable attempt to build a financial model was performed; we’re hoping to simply get a sense of how these solutions compare). The modified ARFE per dollar (including the generated value from the solutions) is shown below for the nominal case for the subset of solutions that deliver positive climate impact.

It is interesting to note that planting food, installing a PV panel, and installing a solar heater have negative ARFE per dollar – this is because they have net negative cost (ignoring the cost of capital)! If we instead invert our analysis to reflect the cost to abate CO₂e, then we end up with, effectively, the marginal abatement cost curve for our assessed solutions: 

One might also be tempted to ask exactly how many square meters of sun-exposed land we might need to drive macroscopic climate change mitigation. Below, the required surface area per person (assuming a population of 7.6 billion) of deployed solution to drive a 1 degree Celsius temperature change over 20 years is shown.

This corresponds to the fraction of the surface area of Earth per the chart below:

For a quick sanity check – the required fraction of Earth’s surface for ‘Plant a Tree’ to reduce temperature vs. business as usual by a degree is ~10% of today’s global forest footprint. We already use more land for agriculture than there is forested land on the planet today, and the amount of managed (i.e. cultivated) agricultural land well-exceeds the fraction of the Earth’s surface that would be required to achieve the 1 degree reduction.

The Conclusion

So… what IS the most impactful way to use a square meter of sun-exposed surface area to mitigate climate change, measured over the next 20 years?

It depends on what one hopes to accomplish. If land use efficiency is more important than cost, then nothing will likely beat the direct approach of removing GHGs from the atmosphere. When taking cost into account, renewable energy generation appears to be a clear winner. When considering extensibility for addressing climate change and total capital deployment requirements, harnessing the power of biology to draw down carbon seems quite sensible.

As always, the goal of these assessments is to provide insight to the key variables that control the outcomes that matter in climate – the important question that remains to be answered by each of us is ‘how do we hope to participate in achieving climate change mitigation?’

References