Dreamtimer

3rd February 2018, 16:05

This is for the math folk.

In his excellent book Physics of Societal Issues: Calculations on National Security, Environment, and Energy, physicist David Hafemeister says that he wants to teach his students to think like the great Italian-American physicist Enrico Fermi. That sounds like a worthy goal, but why Fermi in particular? The answer to that question has two parts.

First, the main premise of Hafemeister’s book is that societal issues should whenever possible be quantified before attempting to formulate possible solutions. Second, a corollary to that premise is that the best way to start the quantification process is to frame the issue as a Fermi problem. But what exactly is a Fermi problem?

The societal issue that will be framed as a Fermi problem is the periodically and recently reported prediction that Earth is headed for a mini ice age in a decade or so, due to an unusually large reduction in the Sun’s energy output around that time. See for example environmental scientist Dana Nuccitelli’s informative article "The 'imminent mini ice age' myth is back, and it's still wrong"

Agricultural scientist Sou, the keeper of the valuable HotWhopper website, did us a considerable favor when she transcribed[4] Radio National New Zealand’s 13 July 2015 interview of Professor Zharkova.[5] Professor Zharkova seemingly just wanted to talk about her solar research, but was repeatedly pressed by the interviewer to opine on what her research meant for human-caused climate change. After demurring several times on the grounds that she was a solar physicist, not a climatologist, she finally ventured a guess. Here is a key quote of Professor Zharkova from the RNZ interview:

… the solar irradiance which comes to the earth will drop to the level that say like the Maunder Minima in 1645-1700, which was about 3 W/m2 reduced irradiance from the sun.

And here is another:

So it will be temperature less irradiance by 3 W/m2 which gives you a decrease of the temperature probably a couple of degrees. I don't know umm.

As it turns out, when it comes to Earth’s surface temperature, Professor Zharkova really doesn’t know, since as we shall see her guess is too big by an order of magnitude, according to those who do know; we are however prepared to accept her figure for the drop in solar irradiance, since that is her field of expertise.

(Note: The Maunder Minimum[6] contributed to, but was neither the initiator nor the sole cause of the so-called Little Ice Age.[7][8])

Equipped with Professor Zharkova’s estimate for the drop in solar irradiance, we are now ready to set up the Fermi problem, or actually two Fermi problems, to boost confidence in the answers.

The first model we shall use is the simple n-layer model of the atmosphere.

In this model, the formula for the average surface temperature T (in units of K = kelvins) of the Earth is:[10]

T = [(n + 1)S (1 - α )/(4σ )]¼ (1)

The quantities appearing in the RHS (right-hand side) of Eqn. (1) are

n = number of atmospheric layers

S = solar constant (in units of W/m2 = watts per square meter)[12]

α = Earth’s albedo (reflectivity to solar radiation)[13]

σ = Stefan–Boltzmann constant = 5.67X10-8 (W/m2)/K4 (K = kelvin = unit of absolute temperature)[14]

As we saw above, according to Professor Zharkova the decrement ΔS in the solar constant S will be 3 W/m2. Now it might be supposed that in order to plug that value into Eqn. (1) to estimate the decrement ΔT in the average surface temperature T we would need to know the putative number of layers n in Earth’s atmosphere; however, that is not the case.

Instead we proceed by defining the sensitivity of T to S as the ratio of the relative change in T to the relative change in S:

Sensitivity = (ΔT/T )/(ΔS/S) (2)

Applying Eqn. (2) to Eqn. (3) (this step requires elementary differential calculus) we find that:

Sensitivity = ¼

This is a very fortunate result since it means that the sensitivity is independent of the value of n.

We can now calculate the value of ΔT as follows:

ΔT = Sensitivity×(T/S )×ΔS = ¼×(288/1360)×(-3) = -0.16 °C

Here we have made use of the following facts[10]

T = 288 K (= 15 °C, just so you know)

S =1360 W/m2 = present value of the solar constant

The Kelvin and Celsius temperature scales use degrees of the same size

But the above value for ΔT doesn't take into account the fast climate feedbacks (i.e., the feedbacks that operate on the century timescale).[10][15] The rule of thumb is that the feedbacks roughly double the effect, so we have:

ΔT ≈ -0.3 °C with feedbacks (retaining only one significant figure)

This value is indeed an order of magnitude lower than Professor Zharkova’s “couple of degrees.”

The second model we use is based on the simplified equations of the WGI (Working Group I) contribution to TAR (Third Assessment Report) of the IPCC (Intergovernmental Panel on Climate Change), which is available both in book form[16] and as a PDF file.[17] The model we need consists of Eqn. (6.1) on p. 354 of TAR WGI:

ΔT = 0.5 ΔF TAR WGI (6.1)

The quantity ΔF in the RHS of the above equation is the change in the radiative forcing F and in this case is the same as the decrement in the solar constant ΔS but divided by 4 to account for the curvature of the Earth’s surface.[10] We therefore calculate as follows:

ΔT = 0.5×(-3)/4 ≈ -0.4 °C (again retaining only one significant figure)

This time we don’t need to double the result because the TAR Eqn. (6.1) already has the effects of the fast climate feedbacks built into it.

How do these solutions to the Fermi problems compare with the numbers obtained by actual climate scientists using more sophisticated approaches? We return to the article by Dana Nuccitelli for an answer:[3]

The most important takeaway point is that the scientific research is clear – were one to occur, a grand solar minimum would temporarily reduce global temperatures by less than 0.3°C...

Agreement is good!

When we combine these results with the following facts[10]

1 °C of warming has already taken place since the advent of the Industrial revolution

There is additional committed warming in the pipeline due to the high thermal inertia of the climate system

The stock of CO2 in the atmosphere is still going up

CO2 once added to the atmosphere stays there for centuries to millennia, whereas

A solar cycle only lasts for about a decade

In his excellent book Physics of Societal Issues: Calculations on National Security, Environment, and Energy, physicist David Hafemeister says that he wants to teach his students to think like the great Italian-American physicist Enrico Fermi. That sounds like a worthy goal, but why Fermi in particular? The answer to that question has two parts.

First, the main premise of Hafemeister’s book is that societal issues should whenever possible be quantified before attempting to formulate possible solutions. Second, a corollary to that premise is that the best way to start the quantification process is to frame the issue as a Fermi problem. But what exactly is a Fermi problem?

The societal issue that will be framed as a Fermi problem is the periodically and recently reported prediction that Earth is headed for a mini ice age in a decade or so, due to an unusually large reduction in the Sun’s energy output around that time. See for example environmental scientist Dana Nuccitelli’s informative article "The 'imminent mini ice age' myth is back, and it's still wrong"

Agricultural scientist Sou, the keeper of the valuable HotWhopper website, did us a considerable favor when she transcribed[4] Radio National New Zealand’s 13 July 2015 interview of Professor Zharkova.[5] Professor Zharkova seemingly just wanted to talk about her solar research, but was repeatedly pressed by the interviewer to opine on what her research meant for human-caused climate change. After demurring several times on the grounds that she was a solar physicist, not a climatologist, she finally ventured a guess. Here is a key quote of Professor Zharkova from the RNZ interview:

… the solar irradiance which comes to the earth will drop to the level that say like the Maunder Minima in 1645-1700, which was about 3 W/m2 reduced irradiance from the sun.

And here is another:

So it will be temperature less irradiance by 3 W/m2 which gives you a decrease of the temperature probably a couple of degrees. I don't know umm.

As it turns out, when it comes to Earth’s surface temperature, Professor Zharkova really doesn’t know, since as we shall see her guess is too big by an order of magnitude, according to those who do know; we are however prepared to accept her figure for the drop in solar irradiance, since that is her field of expertise.

(Note: The Maunder Minimum[6] contributed to, but was neither the initiator nor the sole cause of the so-called Little Ice Age.[7][8])

Equipped with Professor Zharkova’s estimate for the drop in solar irradiance, we are now ready to set up the Fermi problem, or actually two Fermi problems, to boost confidence in the answers.

The first model we shall use is the simple n-layer model of the atmosphere.

In this model, the formula for the average surface temperature T (in units of K = kelvins) of the Earth is:[10]

T = [(n + 1)S (1 - α )/(4σ )]¼ (1)

The quantities appearing in the RHS (right-hand side) of Eqn. (1) are

n = number of atmospheric layers

S = solar constant (in units of W/m2 = watts per square meter)[12]

α = Earth’s albedo (reflectivity to solar radiation)[13]

σ = Stefan–Boltzmann constant = 5.67X10-8 (W/m2)/K4 (K = kelvin = unit of absolute temperature)[14]

As we saw above, according to Professor Zharkova the decrement ΔS in the solar constant S will be 3 W/m2. Now it might be supposed that in order to plug that value into Eqn. (1) to estimate the decrement ΔT in the average surface temperature T we would need to know the putative number of layers n in Earth’s atmosphere; however, that is not the case.

Instead we proceed by defining the sensitivity of T to S as the ratio of the relative change in T to the relative change in S:

Sensitivity = (ΔT/T )/(ΔS/S) (2)

Applying Eqn. (2) to Eqn. (3) (this step requires elementary differential calculus) we find that:

Sensitivity = ¼

This is a very fortunate result since it means that the sensitivity is independent of the value of n.

We can now calculate the value of ΔT as follows:

ΔT = Sensitivity×(T/S )×ΔS = ¼×(288/1360)×(-3) = -0.16 °C

Here we have made use of the following facts[10]

T = 288 K (= 15 °C, just so you know)

S =1360 W/m2 = present value of the solar constant

The Kelvin and Celsius temperature scales use degrees of the same size

But the above value for ΔT doesn't take into account the fast climate feedbacks (i.e., the feedbacks that operate on the century timescale).[10][15] The rule of thumb is that the feedbacks roughly double the effect, so we have:

ΔT ≈ -0.3 °C with feedbacks (retaining only one significant figure)

This value is indeed an order of magnitude lower than Professor Zharkova’s “couple of degrees.”

The second model we use is based on the simplified equations of the WGI (Working Group I) contribution to TAR (Third Assessment Report) of the IPCC (Intergovernmental Panel on Climate Change), which is available both in book form[16] and as a PDF file.[17] The model we need consists of Eqn. (6.1) on p. 354 of TAR WGI:

ΔT = 0.5 ΔF TAR WGI (6.1)

The quantity ΔF in the RHS of the above equation is the change in the radiative forcing F and in this case is the same as the decrement in the solar constant ΔS but divided by 4 to account for the curvature of the Earth’s surface.[10] We therefore calculate as follows:

ΔT = 0.5×(-3)/4 ≈ -0.4 °C (again retaining only one significant figure)

This time we don’t need to double the result because the TAR Eqn. (6.1) already has the effects of the fast climate feedbacks built into it.

How do these solutions to the Fermi problems compare with the numbers obtained by actual climate scientists using more sophisticated approaches? We return to the article by Dana Nuccitelli for an answer:[3]

The most important takeaway point is that the scientific research is clear – were one to occur, a grand solar minimum would temporarily reduce global temperatures by less than 0.3°C...

Agreement is good!

When we combine these results with the following facts[10]

1 °C of warming has already taken place since the advent of the Industrial revolution

There is additional committed warming in the pipeline due to the high thermal inertia of the climate system

The stock of CO2 in the atmosphere is still going up

CO2 once added to the atmosphere stays there for centuries to millennia, whereas

A solar cycle only lasts for about a decade