It looks like the ECS distribution is not normally distributed, even though $B$ is.
👉 How does $\overline{\text{ECS}(B)}$ compare to $\text{ECS}(\overline{B})$? What is the probability that $\text{ECS}(B)$ lies above $\text{ECS}(\overline{B})$?
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Exercise XX:Lecture transcript
(MIT students only)
Please see the link for hw 9 transcript document on Canvas. We want each of you to correct about 500 lines, but don’t spend more than 20 minutes on it. See the the beginning of the document for more instructions. :point_right: Please mention the name of the video(s) and the line ranges you edited:
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Exercise 1.5 - Running the model
In the lecture notebook we introduced a mutable structEBM (energy balance model), which contains:
the parameters of our climate simulation (C, a, A, B, CO2_PI, α, S, see details below)
a function CO2, which maps a time t to the concentrations at that year. For example, we use the function t -> 280 to simulate a model with concentrations fixed at 280 ppm.
EBM also contains the simulation results, in two arrays:
T is the array of tempartures (°C, Float64).
t is the array of timestamps (years, Float64), of the same size as T.
Exercise 1.6 - Application to policy relevant questions
We talked about two emissions scenarios: RCP2.6 (strong mitigation - controlled CO2 concentrations) and RCP8.5 (no mitigation - high CO2 concentrations). These are given by the following functions:
We talked about a second theory – a large increase in CO₂ (by volcanoes) could have caused a strong enough greenhouse effect to melt the Snowball. If we imagine that the CO₂ then decreased (e.g. by getting sequestered by the now liquid ocean), we might be able to explain how we transitioned from a hostile Snowball Earth to today's habitable "Waterball" Earth.
In this exercise, you will estimate how much CO₂ would be needed to melt the Snowball and visualize a possible trajectory for Earth's climate over the past 700 million years by making an interactive bifurcation diagram.
Exercise 2.1
In the lecture notebook (video above), we had a bifurcation diagram of $S$ (solar insolation) vs $T$ (temperature). We increased $S$, watched our point move right in the diagram until we found the tipping point. This time we will do the same, but we vary the CO₂ concentration, and keep $S$ fixed at its default (present day) value.
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Hint
The function findfirst might be helpful.
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Use a condition on the albedo or temperature to check whether the Snowball has melted.
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If you like, make the visualization more informative! Like in the lecture notebook, you could add a trail behind the black dot, or you could plot the stable and unstable branches. It's up to you!
Again, look inside simulated_model and notice that T and t have accumulated the simulation results.
In this simulation, we used T0 = 14 and CO2 = t -> 280, which is why T is constant during our simulation. These parameters are the default, pre-industrial values, and our model is based on this equilibrium.
👉 Run a simulation with policy scenario RCP8.5, and plot the computed temperature graph. What is the global temperature at 2100?
In lecture 21 (see below), we discovered that increases in the brightness of the Sun are not sufficient to explain how Snowball Earth eventually melted.
We are interested in how the uncertainty in our input$B$ (the climate feedback paramter) propagates through our model to determine the uncertainty in our output$T(t)$, for a given emissions scenario. The goal of this exercise is to answer the following by using Monte Carlo Simulation for uncertainty propagation:
👉 What is the probability that we see more than 2°C of warming by 2100 under the low-emissions scenario RCP2.6? What about under the high-emissions scenario RCP8.5?
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👉 Change the value of $B$ using the slider above. What does it mean for a climate system to have a more negative value of $B$? Explain why we call $B$ the climate feedback parameter.
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Before you submit
Remember to fill in your name and Kerberos ID at the top of this notebook.
A recent ground-breaking review paper produced the most comprehensive and up-to-date estimate of the climate feedback parameter, which they find to be
$$B \approx \mathcal{N}(-1.3, 0.4),$$
i.e. our knowledge of the real value is normally distributed with a mean value $\overline{B} = -1.3$ W/m²/K and a standard deviation $\sigma = 0.4$ W/m²/K. These values are not very intuitive, so let us convert them into more policy-relevant numbers.
Definition:Equilibrium climate sensitivity (ECS) is defined as the amount of warming $\Delta T$ caused by a doubling of CO₂ (e.g. from the pre-industrial value 280 ppm to 560 ppm), at equilibrium.
At equilibrium, the energy balance model equation is:
$$0 = \frac{S(1 - α)}{4} - (A - BT_{eq}) + a \ln\left( \frac{2\;\text{CO}₂_{\text{PI}}}{\text{CO}₂_{\text{PI}}} \right)$$
From this, we subtract the preindustrial energy balance, which is given by:
$$0 = \frac{S(1-α)}{4} - (A - BT_{0}),$$
The result of this subtraction, after rearranging, is our definition of $\text{ECS}$:
Before working on the homework, make sure that you have watched the first lecture on climate modeling 👆. We have included the important functions from this lecture notebook in the next cell. Feel free to have a look!
👉 Write a function step_model! that takes an existing ebm and new_CO2, which performs a step of our interactive process:
Reset the model by setting the ebm.t and ebm.T arrays to a single element. Which value?
Assign a new function to ebm.CO2. What function?
Run the model.
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👉 Generate a probability distribution for the ECS based on the probability distribution function for $B$ above. Plot a histogram.
👉 Write a function temperature_response that takes a function CO2 and an optional value B as parameters, and returns the temperature at 2100 according to our model.
To compute ECS, we doubled the CO₂ in our atmosphere. This factor 2 is not entirely arbitrary: without substantial effort to reduce CO₂ emissions, we are expected to at least double the CO₂ in our atmosphere by 2100.
Right now, our CO₂ concentration is 415 ppm – 1.482 times the pre-industrial value of 280 ppm from 1850.
The CO₂ concentrations in the future depend on human action. There are several models for future concentrations, which are formed by assuming different policy scenarios. A baseline model is RCP8.5 - a "worst-case" high-emissions scenario. In our notebook, this model is given as a function of $t$.
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Below we have an empty diagram, which is already set up with a CO₂ vs $T$ diagram, with a logirthmic horizontal axis. Now it's your turn! We have written some pointers below to help you, but feel free to do it your own way.
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Hint
Start by writing a function equilibrium_temperature(CO2) which creates a new EBM at the Snowball Earth temperature T = -48 and returns the final temperature for a given CO2 level.
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👉 What happens when $B$ is greater than or equal to zero?
The climate feedback parameter $B$ is not something that we can control– it is an emergent property of the global climate system. Unfortunately, $B$ is also difficult to quantify empirically (the relevant processes are difficult or impossible to observe directly), so there remains uncertainty as to its exact value.
A value of $B$ close to zero means that an increase in CO₂ concentrations will have a larger impact on global warming, and that more action is needed to stay below a maximum temperature. In answering such policy-related question, we need to take the uncertainty in $B$ into account. In this exercise, we will do so using a Monte Carlo simulation: we generate a sample of values for $B$, and use these values in our analysis.
The plot below provides an example of an "abrupt 2xCO₂" experiment, a classic experimental treatment method in climate modelling which is used in practice to estimate ECS for a particular model. (Note: in complicated climate models the values of the parameters $a$ and $B$ are not specified a priori, but emerge as outputs of the simulation.)
The simulation begins at the preindustrial equilibrium, i.e. a temperature $T_{0} = 14$°C is in balance with the pre-industrial CO₂ concentration of 280 ppm until CO₂ is abruptly doubled from 280 ppm to 560 ppm. The climate responds by warming rapidly, and after a few hundred years approaches the equilibrium climate sensitivity value, by definition.
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cell_id$a86f13de-259d-11eb-3f46-1f6fb40020cedownstream_cells_mapobservations_from_changing_Bupstream_cells_map@md_strgetindex$930d7154-1fbf-11eb-1c3a-b1970d291811precedence_heuristic cell_id$930d7154-1fbf-11eb-1c3a-b1970d291811downstream_cells_mapModel$c4398f9c-1fc4-11eb-0bbb-37f066c6027d$25f92dec-1fc4-11eb-055d-f34deea81d0e$e10a9b70-25a0-11eb-2aed-17ed8221c208$746aa5bc-266c-11eb-14c9-63ccc313f5de$bfb07a0a-2670-11eb-3938-772499c637b1$09901de6-2672-11eb-3d50-05b176b729e7$aea0d0b4-2672-11eb-231e-395c863827d3$19957754-252d-11eb-1e0a-930b5208f5ac$0e19f82e-2685-11eb-2e99-0d094c1aa520$06d28052-2531-11eb-39e2-e9613ab0401c$736515ba-2685-11eb-38cb-65bfcf8d1b8d$d7801e88-2530-11eb-0b93-6f1c78d00eea$607058ec-253c-11eb-0fb6-add8cfb73a4fupstream_cells_map$50ea30ba-25a1-11eb-05d8-b3d579f85652precedence_heuristic cell_id$50ea30ba-25a1-11eb-05d8-b3d579f85652downstream_cells_mapexpected_double_CO2_yearupstream_cells_mapmissing$19957754-252d-11eb-1e0a-930b5208f5acprecedence_heuristic cell_id$19957754-252d-11eb-1e0a-930b5208f5acdownstream_cells_mapupstream_cells_mapModel.CO2_RCP26Model.CO2_RCP85Model$930d7154-1fbf-11eb-1c3a-b1970d291811t_scenario_test$40f1e7d8-252d-11eb-0549-49ca4e806e16$d6d1b312-2543-11eb-1cb2-e5b801686ffbprecedence_heuristic cell_id$d6d1b312-2543-11eb-1cb2-e5b801686ffbdownstream_cells_mapupstream_cells_map@md_strgetindex$02232964-2603-11eb-2c4c-c7b7e5fed7d1precedence_heuristic cell_id$02232964-2603-11eb-2c4c-c7b7e5fed7d1downstream_cells_mapB̅$c4398f9c-1fc4-11eb-0bbb-37f066c6027d$736ed1b6-1fc2-11eb-359e-a1be0a188670σ$736ed1b6-1fc2-11eb-359e-a1be0a188670upstream_cells_map$40f1e7d8-252d-11eb-0549-49ca4e806e16precedence_heuristic cell_id$40f1e7d8-252d-11eb-0549-49ca4e806e16downstream_cells_mapt_scenario_test$19957754-252d-11eb-1e0a-930b5208f5acupstream_cells_mapCoreBasePlutoRunner.create_bondPlutoRunnert$ee1be5dc-252b-11eb-0865-291aa823b9e9Core.applicable@bindBase.getSlider$253f4da0-2433-11eb-1e48-4906059607d3precedence_heuristic cell_id$253f4da0-2433-11eb-1e48-4906059607d3downstream_cells_mapupstream_cells_map@md_strgetindex$d7801e88-2530-11eb-0b93-6f1c78d00eeaprecedence_heuristic cell_id$d7801e88-2530-11eb-0b93-6f1c78d00eeadownstream_cells_mapα$607058ec-253c-11eb-0fb6-add8cfb73a4fupstream_cells_map<=<->=/isless+Model$930d7154-1fbf-11eb-1c3a-b1970d291811*$232b9bec-2544-11eb-0401-97a60bb172fcprecedence_heuristic cell_id$232b9bec-2544-11eb-0401-97a60bb172fcdownstream_cells_mapupstream_cells_map@md_strTneo$de95efae-2675-11eb-0909-73afcd68fd42hint$36f8c1e8-2433-11eb-1f6e-69dc552a4a07getindex$736ed1b6-1fc2-11eb-359e-a1be0a188670precedence_heuristic cell_id$736ed1b6-1fc2-11eb-359e-a1be0a188670downstream_cells_mapB_samples$49cb5174-1fc3-11eb-3670-c3868c9b0255upstream_cells_map 280` to simulate a model with concentrations fixed at 280 ppm.
`EBM` also contains the simulation results, in two arrays:
- `T` is the array of tempartures (°C, `Float64`).
- `t` is the array of timestamps (years, `Float64`), of the same size as `T`.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$049a866e-2672-11eb-29f7-bfea7ad8f572cell_id$049a866e-2672-11eb-29f7-bfea7ad8f572codetemperature_response(t -> 280)metadatashow_logsèdisabled®skip_as_script«code_folded$02173c7a-2695-11eb-251c-65efb5b4a45fcell_id$02173c7a-2695-11eb-251c-65efb5b4a45fcodemetadatashow_logsèdisabled®skip_as_script«code_folded$49cb5174-1fc3-11eb-3670-c3868c9b0255cell_id$49cb5174-1fc3-11eb-3670-c3868c9b0255code\histogram(B_samples, size=(600, 250), label=nothing, xlabel="B [W/m²/K]", ylabel="samples")metadatashow_logsèdisabled®skip_as_script«code_folded$09ce27ca-268c-11eb-0cdd-c9801db876f8cell_id$09ce27ca-268c-11eb-0cdd-c9801db876f8codemd"""
##### Parameters
"""metadatashow_logsèdisabled®skip_as_script«code_folded$53c2eaf6-268b-11eb-0899-b91c03713da4cell_id$53c2eaf6-268b-11eb-0899-b91c03713da4codeRhint(md"
```julia
@bind log_CO2 Slider(❓)
```
```julia
CO2 = 10^log_CO2
```
")metadatashow_logsèdisabled®skip_as_script«code_folded$9c32db5c-1fc9-11eb-029a-d5d554de1067cell_id$9c32db5c-1fc9-11eb-029a-d5d554de1067codemd"""#### Exercise 1.6 - _Application to policy relevant questions_
We talked about two _emissions scenarios_: RCP2.6 (strong mitigation - controlled CO2 concentrations) and RCP8.5 (no mitigation - high CO2 concentrations). These are given by the following functions:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$37552044-2433-11eb-1984-d16e355a7c10cell_id$37552044-2433-11eb-1984-d16e355a7c10codehTODO = html"TODO"metadatashow_logsèdisabled®skip_as_script«code_folded$16348b6a-1fc2-11eb-0b9c-65df528db2a1cell_id$16348b6a-1fc2-11eb-0b9c-65df528db2a1code\md"""
##### Exercise 1.1 - _Develop understanding for feedbacks and climate sensitivity_
"""metadatashow_logsèdisabled®skip_as_script«code_folded$21524c08-2433-11eb-0c55-47b1bdc9e459cell_id$21524c08-2433-11eb-0c55-47b1bdc9e459codeFmd"""
# **Homework 9**: _Climate modeling I_
`18.S191`, fall 2020
"""metadatashow_logsèdisabled®skip_as_script«code_folded$1f148d9a-1fc8-11eb-158e-9d784e390b24cell_id$1f148d9a-1fc8-11eb-158e-9d784e390b24codemetadatashow_logsèdisabled®skip_as_script«code_folded$11096250-2544-11eb-057b-d7112f20b05ccell_id$11096250-2544-11eb-057b-d7112f20b05ccodezmd"""
#### Exercise 2.2
👉 Find the **lowest CO₂ concentration** necessary to melt the Snowball, programatically.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$7d815988-1fc7-11eb-322a-4509e7128ce3cell_id$7d815988-1fc7-11eb-322a-4509e7128ce3codeif reveal_nonnegative_B_answer
md"""
This is known as the "runaway greenhouse effect", where warming self-amplifies so strongly through *positive feedbacks* that the warming continues forever (or until the oceans boil away and there is no longer a reservoir or water to support a *water vapor feedback*. This is thought to explain Venus' extremely hot and hostile climate, but as you can see is extremely unlikely to occur on present-day Earth.
"""
endmetadatashow_logsèdisabled®skip_as_script«code_folded$1e06178a-1fbf-11eb-32b3-61769a79b7c0cell_id$1e06178a-1fbf-11eb-32b3-61769a79b7c0codebegin
import Pkg
Pkg.activate(mktempdir())
Pkg.add([
"Plots",
"PlutoUI",
"LaTeXStrings",
"Distributions",
"Random",
])
using LaTeXStrings
using Plots
using PlutoUI
using Random, Distributions
endmetadatashow_logsèdisabled®skip_as_script«code_folded$169727be-2433-11eb-07ae-ab7976b5be90cell_id$169727be-2433-11eb-07ae-ab7976b5be90codemd"_homework 9, version 1_"metadatashow_logsèdisabled®skip_as_script«code_folded$3e310cf8-25ec-11eb-07da-cb4a2c71ae34cell_id$3e310cf8-25ec-11eb-07da-cb4a2c71ae34code
md"""
We talked about a second theory -- a large increase in CO₂ (by volcanoes) could have caused a strong enough greenhouse effect to melt the Snowball. If we imagine that the CO₂ then decreased (e.g. by getting sequestered by the now liquid ocean), we might be able to explain how we transitioned from a hostile Snowball Earth to today's habitable "Waterball" Earth.
In this exercise, you will estimate how much CO₂ would be needed to melt the Snowball and visualize a possible trajectory for Earth's climate over the past 700 million years by making an interactive *bifurcation diagram*.
#### Exercise 2.1
In the [lecture notebook](https://github.com/hdrake/simplEarth/blob/master/2_ebm_multiple_equilibria.jl) (video above), we had a bifurcation diagram of $S$ (solar insolation) vs $T$ (temperature). We increased $S$, watched our point move right in the diagram until we found the tipping point. This time we will do the same, but we vary the CO₂ concentration, and keep $S$ fixed at its default (present day) value.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$298deff4-2676-11eb-2595-e7e22f613ea1cell_id$298deff4-2676-11eb-2595-e7e22f613ea1codeCO2min = 10metadatashow_logsèdisabled®skip_as_script«code_folded$f688f9f2-2671-11eb-1d71-a57c9817433fcell_id$f688f9f2-2671-11eb-1d71-a57c9817433fcodeSfunction temperature_response(CO2::Function, B::Float64=-1.3)
return missing
endmetadatashow_logsèdisabled®skip_as_script«code_folded$51e2e742-25a1-11eb-2511-ab3434eacc3ecell_id$51e2e742-25a1-11eb-2511-ab3434eacc3ecode4hint(md"The function `findfirst` might be helpful.")metadatashow_logsèdisabled®skip_as_script«code_folded$746aa5bc-266c-11eb-14c9-63ccc313f5decell_id$746aa5bc-266c-11eb-14c9-63ccc313f5decodeqempty_ebm = Model.EBM(
14.0, # initial temperature
1850, # initial year
1, # Δt
t -> 280.0, # CO2 function
)metadatashow_logsèdisabled®skip_as_script«code_folded$68b2a560-2536-11eb-0cc4-27793b4d6a70cell_id$68b2a560-2536-11eb-0cc4-27793b4d6a70codefunction add_cold_hot_areas!(p)
left, right = xlims(p)
plot!(p,
[left, right], [-60, -60],
fillrange=[-10., -10.], fillalpha=0.3, c=:lightblue, label=nothing
)
annotate!(p,
left+12, -19,
text("completely\nfrozen", 10, :darkblue, :left)
)
plot!(p,
[left, right], [10, 10],
fillrange=[80., 80.], fillalpha=0.09, c=:red, lw=0., label=nothing
)
annotate!(p,
left+12, 15,
text("no ice", 10, :darkred, :left)
)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$23335418-2433-11eb-05e4-2b35dc6cca0ecell_id$23335418-2433-11eb-05e4-2b35dc6cca0ecode# edit the code below to set your name and kerberos ID (i.e. email without @mit.edu)
student = (name = "Jazzy Doe", kerberos_id = "jazz")
# you might need to wait until all other cells in this notebook have completed running.
# scroll around the page to see what's upmetadatashow_logsèdisabled®skip_as_script«code_folded$25f92dec-1fc4-11eb-055d-f34deea81d0ecell_id$25f92dec-1fc4-11eb-055d-f34deea81d0ecodelet
double_CO2(t) = if t >= 0
2*Model.CO2_PI
else
Model.CO2_PI
end
# the definition of A depends on B, so we recalculate:
A = Model.S*(1. - Model.α)/4 + B_slider*Model.T0
# create the model
ebm_ECS = Model.EBM(14., -100., 1., double_CO2, A=A, B=B_slider);
Model.run!(ebm_ECS, 300)
ecs = ECS(B=B_slider)
p = plot(
size=(500,250), legend=:bottomright,
title="Transient response to instant doubling of CO₂",
ylabel="temperature change [°C]", xlabel="years after doubling",
ylim=(-.5, (isfinite(ecs) && ecs < 4) ? 4 : 10),
)
plot!(p, [ebm_ECS.t[1], ebm_ECS.t[end]], ecs .* [1,1],
ls=:dash, color=:darkred, label="ECS")
plot!(p, ebm_ECS.t, ebm_ECS.T .- ebm_ECS.T[1],
label="ΔT(t) = T(t) - T₀")
end |> as_svgmetadatashow_logsèdisabled®skip_as_script«code_folded$aea0d0b4-2672-11eb-231e-395c863827d3cell_id$aea0d0b4-2672-11eb-231e-395c863827d3code+temperature_response(Model.CO2_RCP85, -1.0)metadatashow_logsèdisabled®skip_as_script«code_folded$18be4f7c-2433-11eb-33cb-8d90ca6f124ccell_id$18be4f7c-2433-11eb-33cb-8d90ca6f124ccodePmd"""
Submission by: **_$(student.name)_** ($(student.kerberos_id)@mit.edu)
"""metadatashow_logsèdisabled®skip_as_script«code_folded$5041cdee-2527-11eb-154f-0b0c68e11fe3cell_id$5041cdee-2527-11eb-154f-0b0c68e11fe3codeylines_i_edited = md"""
Abstraction, lines 1-219; Array Basics, lines 1-137; Course Intro, lines 1-144 (_for example_)
"""metadatashow_logsèdisabled®skip_as_script«code_folded$378aed18-252b-11eb-0b37-a3b511af2cb5cell_id$378aed18-252b-11eb-0b37-a3b511af2cb5codelet
p = plot(
xlims=(CO2min, CO2max), ylims=(-55, 75),
xaxis=:log,
xlabel="CO2 concentration [ppm]",
ylabel="Global temperature T [°C]",
title="Earth's CO2 concentration bifurcation diagram",
legend=:topleft
)
add_cold_hot_areas!(p)
add_reference_points!(p)
# your code here
plot!(p,
[ebm.CO2(ebm.t[end])], [ebm.T[end]],
label=nothing,
color=:black,
shape=:circle,
)
end |> as_svgmetadatashow_logsèdisabled®skip_as_script«code_folded$b6d7a362-1fc8-11eb-03bc-89464b55c6fccell_id$b6d7a362-1fc8-11eb-03bc-89464b55c6fccodemd"**Answer:**"metadatashow_logsèdisabled®skip_as_script«code_folded$cf276892-25e7-11eb-38f0-03f75c90dd9ecell_id$cf276892-25e7-11eb-38f0-03f75c90dd9ecodeAobservations_from_the_order_of_averaging = md"""
Hello world!
"""metadatashow_logsèdisabled®skip_as_script«code_folded$cb15cd88-25ed-11eb-2be4-f31500a726c8cell_id$cb15cd88-25ed-11eb-2be4-f31500a726c8code`hint(md"Use a condition on the albedo or temperature to check whether the Snowball has melted.")metadatashow_logsèdisabled®skip_as_script«code_folded$37061f1e-2433-11eb-3879-2d31dc70a771cell_id$37061f1e-2433-11eb-3879-2d31dc70a771codeSalmost(text) = Markdown.MD(Markdown.Admonition("warning", "Almost there!", [text]))metadatashow_logsèdisabled®skip_as_script«code_folded$9c1f73e0-268a-11eb-2bf1-216a5d869568cell_id$9c1f73e0-268a-11eb-2bf1-216a5d869568codemd"""
If you like, make the visualization more informative! Like in the lecture notebook, you could add a trail behind the black dot, or you could plot the stable and unstable branches. It's up to you!
"""metadatashow_logsèdisabled®skip_as_script«code_folded$56b68356-2601-11eb-39a9-5f4b8e580b87cell_id$56b68356-2601-11eb-39a9-5f4b8e580b87codeBmd"Reveal answer: $(@bind reveal_nonnegative_B_answer CheckBox())"metadatashow_logsèdisabled®skip_as_script«code_folded$12cbbab0-2671-11eb-2b1f-038c206e84cecell_id$12cbbab0-2671-11eb-2b1f-038c206e84cecodemd"""
Again, look inside `simulated_model` and notice that `T` and `t` have accumulated the simulation results.
In this simulation, we used `T0 = 14` and `CO2 = t -> 280`, which is why `T` is constant during our simulation. These parameters are the default, pre-industrial values, and our model is based on this equilibrium.
👉 Run a simulation with policy scenario RCP8.5, and plot the computed temperature graph. What is the global temperature at 2100?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$9596c2dc-2671-11eb-36b9-c1af7e5f1089cell_id$9596c2dc-2671-11eb-36b9-c1af7e5f1089code*simulated_rcp85_model = let
missing
endmetadatashow_logsèdisabled®skip_as_script«code_folded$f94a1d56-2671-11eb-2cdc-810a9c7a8a5fcell_id$f94a1d56-2671-11eb-2cdc-810a9c7a8a5fcodemetadatashow_logsèdisabled®skip_as_script«code_folded$4b091fac-2672-11eb-0db8-75457788d85ecell_id$4b091fac-2672-11eb-0db8-75457788d85ecodemd"""
Additional parameters can be set using keyword arguments. For example:
```julia
Model.EBM(14, 1850, 1, t -> 280.0; B=-2.0)
```
Creates the same model as before, but with `B = -2.0`.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$1eabe908-268b-11eb-329b-b35160ec951ecell_id$1eabe908-268b-11eb-329b-b35160ec951ecodeٛmd"""
👉 Create a slider for `CO2` between `CO2min` and `CO2max`. Just like the horizontal axis of our plot, we want the slider to be _logarithmic_.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$971f401e-266c-11eb-3104-171ae299ef70cell_id$971f401e-266c-11eb-3104-171ae299ef70code7md"""
You can set up an instance of `EBM` like so:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$de95efae-2675-11eb-0909-73afcd68fd42cell_id$de95efae-2675-11eb-0909-73afcd68fd42codeTneo = -48metadatashow_logsèdisabled®skip_as_script«code_folded$2dfab366-25a1-11eb-15c9-b3dd9cd6b96ccell_id$2dfab366-25a1-11eb-15c9-b3dd9cd6b96ccodermd"""
👉 In what year are we expected to have doubled the CO₂ concentration, under policy scenario RCP8.5?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$1ea81214-1fca-11eb-2442-7b0b448b49d6cell_id$1ea81214-1fca-11eb-2442-7b0b448b49d6codemd"""
## **Exercise 2** - _How did Snowball Earth melt?_
In lecture 21 (see below), we discovered that increases in the brightness of the Sun are not sufficient to explain how Snowball Earth eventually melted.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$a919d584-2670-11eb-1cf9-2327c8135d6dcell_id$a919d584-2670-11eb-1cf9-2327c8135d6dcode{md"""
Have look inside this object. We see that `T` and `t` are initialized to a 1-element array.
Let's run our model:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$06c5139e-252d-11eb-2645-8b324b24c405cell_id$06c5139e-252d-11eb-2645-8b324b24c405code md"""
We are interested in how the **uncertainty in our input** $B$ (the climate feedback paramter) *propagates* through our model to determine the **uncertainty in our output** $T(t)$, for a given emissions scenario. The goal of this exercise is to answer the following by using *Monte Carlo Simulation* for *uncertainty propagation*:
> 👉 What is the probability that we see more than 2°C of warming by 2100 under the low-emissions scenario RCP2.6? What about under the high-emissions scenario RCP8.5?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$b9f882d8-266b-11eb-2998-75d6539088c7cell_id$b9f882d8-266b-11eb-2998-75d6539088c7codemetadatashow_logsèdisabled®skip_as_script«code_folded$e296c6e8-259c-11eb-1385-53f757f4d585cell_id$e296c6e8-259c-11eb-1385-53f757f4d585codemd"""
👉 Change the value of $B$ using the slider above. What does it mean for a climate system to have a more negative value of $B$? Explain why we call $B$ the _climate feedback parameter_.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$06d28052-2531-11eb-39e2-e9613ab0401ccell_id$06d28052-2531-11eb-39e2-e9613ab0401ccode.ebm = Model.EBM(Tneo, 0., 5., Model.CO2_const)metadatashow_logsèdisabled®skip_as_script«code_folded$e10a9b70-25a0-11eb-2aed-17ed8221c208cell_id$e10a9b70-25a0-11eb-2aed-17ed8221c208codeOplot(t, Model.CO2_RCP85.(t),
ylim=(0,1200), ylabel="CO2 concentration [ppm]")metadatashow_logsèdisabled®skip_as_script«code_folded$36e2dfea-2433-11eb-1c90-bb93ab25b33ccell_id$36e2dfea-2433-11eb-1c90-bb93ab25b33ccodeif student.name == "Jazzy Doe" || student.kerberos_id == "jazz"
md"""
!!! danger "Before you submit"
Remember to fill in your **name** and **Kerberos ID** at the top of this notebook.
"""
endmetadatashow_logsèdisabled®skip_as_script«code_folded$1312525c-1fc0-11eb-2756-5bc3101d2260cell_id$1312525c-1fc0-11eb-2756-5bc3101d2260codemd"""## **Exercise 1** - _policy goals under uncertainty_
A recent ground-breaking [review paper](https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2019RG000678) produced the most comprehensive and up-to-date estimate of the *climate feedback parameter*, which they find to be
$B \approx \mathcal{N}(-1.3, 0.4),$
i.e. our knowledge of the real value is normally distributed with a mean value $\overline{B} = -1.3$ W/m²/K and a standard deviation $\sigma = 0.4$ W/m²/K. These values are not very intuitive, so let us convert them into more policy-relevant numbers.
**Definition:** *Equilibrium climate sensitivity (ECS)* is defined as the amount of warming $\Delta T$ caused by a doubling of CO₂ (e.g. from the pre-industrial value 280 ppm to 560 ppm), at equilibrium.
At equilibrium, the energy balance model equation is:
$0 = \frac{S(1 - α)}{4} - (A - BT_{eq}) + a \ln\left( \frac{2\;\text{CO}₂_{\text{PI}}}{\text{CO}₂_{\text{PI}}} \right)$
From this, we subtract the preindustrial energy balance, which is given by:
$0 = \frac{S(1-α)}{4} - (A - BT_{0}),$
The result of this subtraction, after rearranging, is our definition of $\text{ECS}$:
$\text{ECS} \equiv T_{eq} - T_{0} = -\frac{a\ln(2)}{B}$
"""metadatashow_logsèdisabled®skip_as_script«code_folded$0e19f82e-2685-11eb-2e99-0d094c1aa520cell_id$0e19f82e-2685-11eb-2e99-0d094c1aa520codefunction add_reference_points!(p)
plot!(p,
[Model.CO2_PI, Model.CO2_PI], [-55, 75],
color=:grey, alpha=0.3, lw=8,
label="Pre-industrial CO2"
)
plot!(p,
[Model.CO2_PI], [Model.T0],
shape=:circle, color=:orange, markersize=8,
label="Our preindustrial climate"
)
plot!(p,
[Model.CO2_PI], [-38.3],
shape=:circle, color=:aqua, markersize=8,
label="Alternate preindustrial climate"
)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$a0ef04b0-25e9-11eb-1110-cde93601f712cell_id$a0ef04b0-25e9-11eb-1110-cde93601f712codehtml"""
"""metadatashow_logsèdisabled®skip_as_script«code_folded$3f823490-266d-11eb-1ba4-d5a23975c335cell_id$3f823490-266d-11eb-1ba4-d5a23975c335codehtml"""
Atmosphere and upper-ocean heat capacity [J/m²/°C]
a
CO₂ forcing effect [W/m²]
CO2_PI
Pre-industrial CO₂ concentration [ppm]
"""metadatashow_logsèdisabled®skip_as_script«code_folded$fe3304f8-2668-11eb-066d-fdacadce5a19cell_id$fe3304f8-2668-11eb-066d-fdacadce5a19codemd"""
_Before working on the homework, make sure that you have watched the first lecture on climate modeling 👆. We have included the important functions from this lecture notebook in the next cell. Feel free to have a look!_
"""metadatashow_logsèdisabled®skip_as_script«code_folded$4c9173ac-2685-11eb-2129-99071821ebebcell_id$4c9173ac-2685-11eb-2129-99071821ebebcode3md"""
👉 Write a function `step_model!` that takes an existing `ebm` and `new_CO2`, which performs a step of our interactive process:
- Reset the model by setting the `ebm.t` and `ebm.T` arrays to a single element. _Which value?_
- Assign a new function to `ebm.CO2`. _What function?_
- Run the model.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$3737be8e-2433-11eb-2049-2d6d8a5e4753cell_id$3737be8e-2433-11eb-2049-2d6d8a5e4753codeYcorrect(text=rand(yays)) = Markdown.MD(Markdown.Admonition("correct", "Got it!", [text]))metadatashow_logsèdisabled®skip_as_script«code_folded$bfb07a0a-2670-11eb-3938-772499c637b1cell_id$bfb07a0a-2670-11eb-3938-772499c637b1codeasimulated_model = let
ebm = Model.EBM(14.0, 1850, 1, t -> 280.0)
Model.run!(ebm, 2020)
ebm
endmetadatashow_logsèdisabled®skip_as_script«code_folded$ee1be5dc-252b-11eb-0865-291aa823b9e9cell_id$ee1be5dc-252b-11eb-0865-291aa823b9e9codet = 1850:2100metadatashow_logsèdisabled®skip_as_script«code_folded$c4398f9c-1fc4-11eb-0bbb-37f066c6027dcell_id$c4398f9c-1fc4-11eb-0bbb-37f066c6027dcode(ECS(; B=B̅, a=Model.a) = -a*log(2.)./B;metadatashow_logsèdisabled®skip_as_script«code_folded$f3abc83c-1fc7-11eb-1aa8-01ce67c8bddecell_id$f3abc83c-1fc7-11eb-1aa8-01ce67c8bddecodeَmd"""
👉 Generate a probability distribution for the ECS based on the probability distribution function for $B$ above. Plot a histogram.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$c78e02b4-268a-11eb-0af7-f7c7620fcc34cell_id$c78e02b4-268a-11eb-0af7-f7c7620fcc34codeBmd"""
The albedo feedback is implemented by the methods below:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$9cdc5f84-2671-11eb-3c78-e3495bc64d33cell_id$9cdc5f84-2671-11eb-3c78-e3495bc64d33codeٷmd"""
👉 Write a function `temperature_response` that takes a function `CO2` and an optional value `B` as parameters, and returns the temperature at 2100 according to our model.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$3cbc95ba-2685-11eb-3810-3bf38aa33231cell_id$3cbc95ba-2685-11eb-3810-3bf38aa33231code'md"""
We used two helper functions:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$aed8f00e-266b-11eb-156d-8bb09de0dc2bcell_id$aed8f00e-266b-11eb-156d-8bb09de0dc2bcodeCmd"""
👉 Create a graph to visualize ECS as a function of B.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$269200ec-259f-11eb-353b-0b73523ef71acell_id$269200ec-259f-11eb-353b-0b73523ef71acodemd"""
#### Exercise 1.2 - _Doubling CO₂_
To compute ECS, we doubled the CO₂ in our atmosphere. This factor 2 is not entirely arbitrary: without substantial effort to reduce CO₂ emissions, we are expected to **at least** double the CO₂ in our atmosphere by 2100.
Right now, our CO₂ concentration is 415 ppm -- $(round(415 / 280, digits=3)) times the pre-industrial value of 280 ppm from 1850.
The CO₂ concentrations in the _future_ depend on human action. There are several models for future concentrations, which are formed by assuming different _policy scenarios_. A baseline model is RCP8.5 - a "worst-case" high-emissions scenario. In our notebook, this model is given as a function of ``t``.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$1d388372-2695-11eb-3068-7b28a2ccb9accell_id$1d388372-2695-11eb-3068-7b28a2ccb9accodemetadatashow_logsèdisabled®skip_as_script«code_folded$8b06b944-268c-11eb-0bfc-8d4dd21e1f02cell_id$8b06b944-268c-11eb-0bfc-8d4dd21e1f02codeEmd"""
👉 Inside the plot cell, call the function `step_model!`.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$372c1480-2433-11eb-3c4e-95a37d51835fcell_id$372c1480-2433-11eb-3c4e-95a37d51835fcodeyays = [md"Fantastic!", md"Splendid!", md"Great!", md"Yay ❤", md"Great! 🎉", md"Well done!", md"Keep it up!", md"Good job!", md"Awesome!", md"You got the right answer!", md"Let's move on to the next section."]metadatashow_logsèdisabled®skip_as_script«code_folded$5f82dec8-259e-11eb-2f4f-4d661f44ef41cell_id$5f82dec8-259e-11eb-2f4f-4d661f44ef41code8observations_from_nonnegative_B = md"""
Hello world!
"""metadatashow_logsèdisabled®skip_as_script«code_folded$fa7e6f7e-2434-11eb-1e61-1b1858bb0988cell_id$fa7e6f7e-2434-11eb-1e61-1b1858bb0988codeWmd"""
``B = `` $(@bind B_slider Slider(-2.5:.001:0; show_value=true, default=-1.3))
"""metadatashow_logsèdisabled®skip_as_script«code_folded$a86f13de-259d-11eb-3f46-1f6fb40020cecell_id$a86f13de-259d-11eb-3f46-1f6fb40020cecode5observations_from_changing_B = md"""
Hello world!
"""metadatashow_logsèdisabled®skip_as_script«code_folded$930d7154-1fbf-11eb-1c3a-b1970d291811cell_id$930d7154-1fbf-11eb-1c3a-b1970d291811code Ymodule Model
const S = 1368; # solar insolation [W/m^2] (energy per unit time per unit area)
const α = 0.3; # albedo, or planetary reflectivity [unitless]
const B = -1.3; # climate feedback parameter [W/m^2/°C],
const T0 = 14.; # preindustrial temperature [°C]
absorbed_solar_radiation(; α=α, S=S) = S*(1 - α)/4; # [W/m^2]
outgoing_thermal_radiation(T; A=A, B=B) = A - B*T;
const A = S*(1. - α)/4 + B*T0; # [W/m^2].
greenhouse_effect(CO2; a=a, CO2_PI=CO2_PI) = a*log(CO2/CO2_PI);
const a = 5.0; # CO2 forcing coefficient [W/m^2]
const CO2_PI = 280.; # preindustrial CO2 concentration [parts per million; ppm];
CO2_const(t) = CO2_PI; # constant CO2 concentrations
const C = 51.; # atmosphere and upper-ocean heat capacity [J/m^2/°C]
function timestep!(ebm)
append!(ebm.T, ebm.T[end] + ebm.Δt*tendency(ebm));
append!(ebm.t, ebm.t[end] + ebm.Δt);
end;
tendency(ebm) = (1. /ebm.C) * (
+ absorbed_solar_radiation(α=ebm.α, S=ebm.S)
- outgoing_thermal_radiation(ebm.T[end], A=ebm.A, B=ebm.B)
+ greenhouse_effect(ebm.CO2(ebm.t[end]), a=ebm.a, CO2_PI=ebm.CO2_PI)
);
begin
mutable struct EBM
T::Array{Float64, 1}
t::Array{Float64, 1}
Δt::Float64
CO2::Function
C::Float64
a::Float64
A::Float64
B::Float64
CO2_PI::Float64
α::Float64
S::Float64
end;
# Make constant parameters optional kwargs
EBM(T::Array{Float64, 1}, t::Array{Float64, 1}, Δt::Real, CO2::Function;
C=C, a=a, A=A, B=B, CO2_PI=CO2_PI, α=α, S=S) = (
EBM(T, t, Δt, CO2, C, a, A, B, CO2_PI, α, S)
);
# Construct from float inputs for convenience
EBM(T0::Real, t0::Real, Δt::Real, CO2::Function;
C=C, a=a, A=A, B=B, CO2_PI=CO2_PI, α=α, S=S) = (
EBM(Float64[T0], Float64[t0], Δt, CO2;
C=C, a=a, A=A, B=B, CO2_PI=CO2_PI, α=α, S=S);
);
end;
begin
function run!(ebm::EBM, end_year::Real)
while ebm.t[end] < end_year
timestep!(ebm)
end
end;
run!(ebm) = run!(ebm, 200.) # run for 200 years by default
end
CO2_hist(t) = CO2_PI * (1 .+ fractional_increase(t));
fractional_increase(t) = ((t .- 1850.)/220).^3;
begin
CO2_RCP26(t) = CO2_PI * (1 .+ fractional_increase(t) .* min.(1., exp.(-((t .-1850.).-170)/100))) ;
RCP26 = EBM(T0, 1850., 1., CO2_RCP26)
run!(RCP26, 2100.)
CO2_RCP85(t) = CO2_PI * (1 .+ fractional_increase(t) .* max.(1., exp.(((t .-1850.).-170)/100)));
RCP85 = EBM(T0, 1850., 1., CO2_RCP85)
run!(RCP85, 2100.)
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$50ea30ba-25a1-11eb-05d8-b3d579f85652cell_id$50ea30ba-25a1-11eb-05d8-b3d579f85652code/expected_double_CO2_year = let
missing
endmetadatashow_logsèdisabled®skip_as_script«code_folded$19957754-252d-11eb-1e0a-930b5208f5accell_id$19957754-252d-11eb-1e0a-930b5208f5accodeBModel.CO2_RCP26(t_scenario_test), Model.CO2_RCP85(t_scenario_test)metadatashow_logsèdisabled®skip_as_script«code_folded$d6d1b312-2543-11eb-1cb2-e5b801686ffbcell_id$d6d1b312-2543-11eb-1cb2-e5b801686ffbcodemd"""
Below we have an empty diagram, which is already set up with a CO₂ vs $T$ diagram, with a logirthmic horizontal axis. Now it's your turn! We have written some pointers below to help you, but feel free to do it your own way.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$02232964-2603-11eb-2c4c-c7b7e5fed7d1cell_id$02232964-2603-11eb-2c4c-c7b7e5fed7d1codeB̅ = -1.3; σ = 0.4metadatashow_logsèdisabled®skip_as_script«code_folded$40f1e7d8-252d-11eb-0549-49ca4e806e16cell_id$40f1e7d8-252d-11eb-0549-49ca4e806e16code>@bind t_scenario_test Slider(t; show_value=true, default=1850)metadatashow_logsèdisabled®skip_as_script«code_folded$253f4da0-2433-11eb-1e48-4906059607d3cell_id$253f4da0-2433-11eb-1e48-4906059607d3code)md"_Let's create a package environment:_"metadatashow_logsèdisabled®skip_as_script«code_folded$d7801e88-2530-11eb-0b93-6f1c78d00eeacell_id$d7801e88-2530-11eb-0b93-6f1c78d00eeacodeٯfunction α(T; α0=Model.α, αi=0.5, ΔT=10.)
if T < -ΔT
return αi
elseif -ΔT <= T < ΔT
return αi + (α0-αi)*(T+ΔT)/(2ΔT)
elseif T >= ΔT
return α0
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$232b9bec-2544-11eb-0401-97a60bb172fccell_id$232b9bec-2544-11eb-0401-97a60bb172fccodehint(md"Start by writing a function `equilibrium_temperature(CO2)` which creates a new `EBM` at the Snowball Earth temperature T = $(Tneo) and returns the final temperature for a given CO2 level.")metadatashow_logsèdisabled®skip_as_script«code_folded$736ed1b6-1fc2-11eb-359e-a1be0a188670cell_id$736ed1b6-1fc2-11eb-359e-a1be0a188670codeٶB_samples = let
B_distribution = Normal(B̅, σ)
Nsamples = 5000
samples = rand(B_distribution, Nsamples)
# we only sample negative values of B
filter(x -> x < 0, samples)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$440271b6-25e8-11eb-26ce-1b80aa176acacell_id$440271b6-25e8-11eb-26ce-1b80aa176acacodeٛmd"👉 Does accounting for uncertainty in feedbacks make our expectation of global warming better (less implied warming) or worse (more implied warming)?"metadatashow_logsèdisabled®skip_as_script«code_folded$36ea4410-2433-11eb-1d98-ab4016245d95cell_id$36ea4410-2433-11eb-1d98-ab4016245d95codeImd"## Function library
Just some helper functions used in the notebook."metadatashow_logsèdisabled®skip_as_script«code_folded$374522c4-2433-11eb-3da3-17419949defccell_id$374522c4-2433-11eb-3da3-17419949defccodeٱnot_defined(variable_name) = Markdown.MD(Markdown.Admonition("danger", "Oopsie!", [md"Make sure that you define a variable called **$(Markdown.Code(string(variable_name)))**"]))metadatashow_logsèdisabled®skip_as_script«code_folded$3d66bd30-259d-11eb-2694-471fb3a4a7becell_id$3d66bd30-259d-11eb-2694-471fb3a4a7becodeFmd"""
👉 What happens when $B$ is greater than or equal to zero?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$bade1372-25a1-11eb-35f4-4b43d4e8d156cell_id$bade1372-25a1-11eb-35f4-4b43d4e8d156code&md"""
#### Exercise 1.3 - _Uncertainty in B_
The climate feedback parameter ``B`` is not something that we can control– it is an emergent property of the global climate system. Unfortunately, ``B`` is also difficult to quantify empirically (the relevant processes are difficult or impossible to observe directly), so there remains uncertainty as to its exact value.
A value of ``B`` close to zero means that an increase in CO₂ concentrations will have a larger impact on global warming, and that more action is needed to stay below a maximum temperature. In answering such policy-related question, we need to take the uncertainty in ``B`` into account. In this exercise, we will do so using a Monte Carlo simulation: we generate a sample of values for ``B``, and use these values in our analysis.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$87e68a4a-2433-11eb-3e9d-21675850ed71cell_id$87e68a4a-2433-11eb-3e9d-21675850ed71codehtml"""
"""metadatashow_logsèdisabled®skip_as_script«code_folded$7f961bc0-1fc5-11eb-1f18-612aeff0d8dfcell_id$7f961bc0-1fc5-11eb-1f18-612aeff0d8dfcodemd"""The plot below provides an example of an "abrupt 2xCO₂" experiment, a classic experimental treatment method in climate modelling which is used in practice to estimate ECS for a particular model. (Note: in complicated climate models the values of the parameters $a$ and $B$ are not specified *a priori*, but *emerge* as outputs of the simulation.)
The simulation begins at the preindustrial equilibrium, i.e. a temperature $T_{0} = 14$°C is in balance with the pre-industrial CO₂ concentration of 280 ppm until CO₂ is abruptly doubled from 280 ppm to 560 ppm. The climate responds by warming rapidly, and after a few hundred years approaches the equilibrium climate sensitivity value, by definition.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$3d72ab3a-2689-11eb-360d-9b3d829b78a9cell_id$3d72ab3a-2689-11eb-360d-9b3d829b78a9codeECS_samples = missingmetadatashow_logsèdisabled®skip_as_script«code_folded«notebook_id$8ddbb042-4aa5-11f0-152c-778cb89daca8in_temp_dir¨metadata