Geo-engineering By Reforesting

Let's imagine we can reduce global carbon emissions to net zero: we still need to make the planet colder to stop the positive feedback in the Arctic. There are various geo-engineering solutions that might be deployed to reduce the temperature of our overheating Earth. None of them are easy or cheap but, given the positive feedback we've unleashed, we desperately need to get something working.

One low-tech idea (published in https://www.science.org/doi/10.1126/science.aax0848) is to plant trees on a massive scale to remove CO₂ from the atmosphere. https://climate.nasa.gov/news/2927/examining-the-viability-of-planting-trees-to-help-mitigate-climate-change/ raises various questions about this approach. For example, planting trees is likely to reduce the land's albedo, meaning less sunlight is reflected, so they may actually increase global temperatures!

The paper suggests that planting trees on 9 million km² of currently unforested land would absorb 25% of current atmospheric CO₂. The global average atmospheric carbon dioxide in 2019 was 409.8 ppm. A 25% reduction would bring that down to 307 ppm, close to the pre-industrial level of 280 ppm, so might be expected to reduce global temperatures by around 1°C - but see Also worth noting for a big caveat.

The area required is about the same as the Sahara. Let's suppose we were able to replace 9 million km² of desert near the equator with trees and check the effect on the Earth's temperature from the change in albedo.

As per A Farewell to Ice p47-49, let's approximate the Earth as a sphere with constant temperature over its whole surface area and use the Stefan-Boltzmann equation for the radiation emitted per unit of surface area of any body. As the Earth is in equilibrium, equate it to the total power the Earth receives from the Sun:
\begin{align} \text{Power emitted as radiation} & = \text{Power received from the Sun} \\ 4{\pi}R^2{\epsilon}{\sigma}{T_0}^4 & = {\pi}R^2S(1-\alpha_{av}) \end{align} where

$R$ is the average radius of the Earth, 6.371 × 10⁶m
$\epsilon$ is the Earth's emissivity; for a planet with no atmosphere $\epsilon$ = 1; according to https://en.wikipedia.org/wiki/Climate_model, with our current atmosphere the Earth's emissivity $\epsilon$ = 0.612
$\sigma$ is the Boltzmann constant, 5.67 × 10^-8W/m²/K⁴
$T_0$ is the Earth's average temperature, around 15°C or 288K
$S$ is the solar constant, 1,370W/m², the amount of radiation reaching the Earth per unit area measured at right angles to the Sun's rays
$\alpha_{av}$ is the Earth's (average) albedo, about 0.30

Now let's suppose we cover area A of the Earth's surface near the equator with trees. For simplicity of visualisation, let's imagine the trees in a band all the away around the Earth near the equator.

Looking at the Earth from the direction of the Sun, the projected area of forest visible at any one time is $$A_{proj} = \frac{A}{\pi}$$

This area of trees will cause less power to be reflected/radiated from the Earth, as follows:
\begin{align} \Delta_P & = \text{power reflected without trees} - \text{power reflected with trees} \\ & = {A_{proj}}{P_s}\alpha_s - {A_{proj}}{P_s}\alpha_t \end{align} where

$P_s$ is the power of raw sunshine reaching the Earth's surface at midday on a cloudless day, 1,000W/m², according to https://www.withouthotair.com/c6/page_38.shtml - less than $S$ because some is absorbed by the atmosphere
$\alpha_t$ is the albedo of deciduous trees (using the middle of the range), 0.165, according to https://forum.corona-renderer.com/index.php?topic=2359.0
$\alpha_s$ is the albedo of desert sand, 0.4, ibid
NB1: for simplicity assume the trees are located somewhere with near-zero cloud cover
NB2: this equilibrium underestimates the warming effect, because it overlooks the reflected sunlight that is absorbed on its way back out of the atmosphere

This simplifies to: $$\Delta_P = \frac{A{P_s}(\alpha_s - \alpha_t)}{\pi}$$

Let's insert this delta into our initial equilibrium equation:
\begin{align} \text{Power emitted as radiation at temperature } T_t - \text{decrease in power reflected} & = \text{Power received from the Sun} \\ 4{\pi}R^2{\epsilon}{\sigma}{T_t}^4 - \frac{A{P_s}(\alpha_s - \alpha_t)}{\pi} & = {\pi}R^2S(1-\alpha_{av}) \end{align} where

$T_t$ is the new (higher) equilibrium, average temperature of the Earth with the trees added

We can substitute the right side of the equation from our initial equilibrium equation: $$4{\pi}R^2{\epsilon}{\sigma}{T_t}^4 - \frac{A{P_s}(\alpha_s - \alpha_t)}{\pi} = 4{\pi}R^2{\epsilon}{\sigma}{T_0}^4$$

Rearranging we get $${T_t} = {\left({T_0}^4 + \frac{A{P_s}(\alpha_s - \alpha_t)}{4{\pi}^2R^2{\epsilon}{\sigma}}\right)}^{1/4}$$

In this case A = 9 x 10¹²m². So $${T_t} = 288.397K$$

That's an increase of around 0.4°C. Uh oh!

What if we reforested not desert but degraded land, partly grass, partly bare earth? In that case, we should replace $\alpha_s$, the albedo of desert sand, in the calculation with:

$\alpha_g$, 0.21, an estimated value between the albedo of bare soil, 0.17, and green grass, 0.25, ibid

The formula above becomes $${T_{tg}} = {\left({T_0}^4 + \frac{A{P_s}(\alpha_g - \alpha_t)}{4{\pi}^2R^2{\epsilon}{\sigma}}\right)}^{1/4}$$ $${T_{tg}} = 288.076K$$

That's an increase of less than 0.1°C.

This means:

We'd be better off planting trees on land that is less reflective than desert sand.
If we reforest desert, all other things being equal, a significantly larger area will need to be reforested to achieve 1°C of temperature reduction.
Where we are reforesting desert, supposing a tree takes 100 years to grow to maturity and absorb the amount of CO₂ expected in the paper, then, from when it and any understorey have grown sufficiently to shade most of the underlying sand until it's perhaps 40 years old, it's likely to cause higher temperatures!

Of course, all things will not be equal. If this project were implemented, it is likely to increase cloud cover overhead which would be expected to decrease temperatures. There are likely to be other side-effects on temperature both positive and negative.

Also worth noting:

The paper is carefully worded that this "change has the potential to store an equivalent of 25% of the current atmospheric carbon pool"
Approximately 50% of the additional CO₂ we release is absorbed by increased biomass and by dissolving into the oceans, as per https://sos.noaa.gov/catalog/datasets/ocean-atmosphere-co2-exchange/
If we reduce atmospheric CO₂ levels, this equilibrium is likely to move into reverse with a net loss of CO₂ from biomass and the oceans
Planting trees on 9 million km² of currently unforested land would reduce atmospheric current CO₂ levels by only half of 25%

Many thanks to my friend Stephen Hartley for bringing to my attention and explaining the opportunity of large-scale reforestation.

Some examples of re-greening:

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