ATM 623: Climate Modeling

Brian E. J. Rose, University at Albany

Lecture 1: Climate models, the global energy budget and Fun with Python

Warning: content out of date and not maintained

You really should be looking at The Climate Laboratory book by Brian Rose, where all the same content (and more!) is kept up to date.

Here you are likely to find broken links and broken code.

About these notes:

This document uses the interactive Jupyter notebook format. The notes can be accessed in several different ways:

• The interactive notebooks are hosted on github at https://github.com/brian-rose/ClimateModeling_courseware
• The latest versions can be viewed as static web pages rendered on nbviewer
• A complete snapshot of the notes as of May 2017 (end of spring semester) are available on Brian's website.

Also here is a legacy version from 2015.

Many of these notes make use of the climlab package, available at https://github.com/brian-rose/climlab

Contents

1. What is a Climate Model?
2. The observed global energy budget
3. Quantifying the planetary energy budget
4. Using Python to compute emission to space

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1. What is a Climate Model?

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First, some thoughts on modeling from xkcd

Let's be a little pedantic and decompose that question:

• what is Climate?
• what is a Model?

Climate is

• statistics of weather, e.g. space and time averages of temperature and precip.
• (statistics might also mean higher-order stats: variability etc)

A model is

• not easy to define!

Wikipedia: http://en.wikipedia.org/wiki/Conceptual_model

In the most general sense, a model is anything used in any way to represent anything else. Some models are physical objects, for instance, a toy model which may be assembled, and may even be made to work like the object it represents. Whereas, a conceptual model is a model made of the composition of concepts, that thus exists only in the mind. Conceptual models are used to help us know, understand, or simulate the subject matter they represent.

George E. P. Box (statistician):

Essentially, all models are wrong, but some are useful.”

From the Climate Modelling Primer, 4th ed (McGuffie and Henderson-Sellers):

In the broadest sense, models are for learning about the world (in our case, the climate) and the learning takes place in the contruction and the manipulation of the model, as anyone who has watched a child build idealised houses or spaceships with Lego, or built with it themselves, will know. Climate models are, likewise, idealised representations of a complicated and complex reality through which our understanding of the climate has significantly expanded. All models involve some ignoring, distoring and approximating, but gradually they allow us to build understanding of the system being modelled. A child's Lego construction typically contains the essential elements of the real objects, improves with attention to detail, helps them understand the real world, but is never confused with the real thing.

A minimal definition of a climate model

A representation of the exchange of energy between the Earth system and space, and its effects on average surface temperature.

(what average?)

Note the focus on planetary energy budget. That’s the key to all climate modeling.

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2. The observed global energy budget

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The figure below shows current best estimates of the global, annual mean energy fluxes through the climate system.

We will look at many of these processes in detail throughout the course.

Things to note:

On the shortwave side

• global mean albedo is 101.9 W m$^{-2}$ / 341.3 W m$^{-2}$ = 0.299
• Reflection off clouds = 79 W m$^{-2}$
• Off surface = 23 W m$^{-2}$
  • 3 times as much reflection off clouds as off surface

Why?? Think about both areas of ice and snow, and the fact that sunlight has to travel through cloudy atmosphere to get to the ice and snow. Also there is some absorption of shortwave by the atmosphere.

• Atmospheric absorption = 78 W m$^{-2}$ (so about the same as reflected by clouds)

QUESTION: Which gases contribute to shortwave absorption?

• O$_3$ and H$_2$O mostly.
• We will look at this later.

On the longwave side

• Observed emission from the SURFACE is 396 W m$^{-2}$
• very close to the blackbody emission $\sigma T^4$ at $T = 288$ K (the global mean surface temperature).
• BUT emission to space is much smaller = 239 W m$^{-2}$

QUESTION: What do we call this? (greenhouse effect)

Look at net numbers…

• Net absorbed = 0.9 W m$^{-2}$
• Why?
• Where is that heat going?

Note, the exchanges of energy between the surface and the atmosphere are complicated, involve a number of different processes. We will look at these more carefully later.

Additional points:

• Notice that this is a budget of energy, not temperature.
• We will need to discuss the connection between the two
• Clouds affect both longwave and shortwave sides of the budget.

• WATER is involved in many of the terms:

  • evaporation
  • latent heating (equal and opposite in the global mean)
  • clouds
  • greenhouse effect
  • atmospheric SW absorption
  • surface reflectivity (ice and snow)

Discussion point

How might we expect some of the terms in the global energy budget to vary under anthropogenic climate change?

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3. Quantifying the planetary energy budget

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A budget for the energy content of the global atmosphere-ocean system:

\begin{align} \frac{dE}{dt} &= \text{net energy flux in to system} \\ &= \text{flux in – flux out} \end{align}

where $E$ is the enthalpy or heat content of the total system.

We will express the budget per unit surface area, so each term above has units W m$^{-2}$

Note: any internal exchanges of energy between different reservoirs (e.g. between ocean, land, ice, atmosphere) do not appear in this budget – because $E$ is the sum of all reservoirs.

Assumption:

The only quantitatively important energy sources to the whole system are radiative fluxes to and from space.

Let’s model those TOA (top-of-atmosphere) fluxes.

Flux in is incoming solar radiation The solar constant is

$$ S_0 = 1365.2 \text{ W m}^{-2} $$

(all values will be consistent with Trenberth and Fasullo figure unless noted otherwise)

This is the flux of energy from the sun incident on a unit area perpendicular to the beam direction.

The area-weighted global mean incoming solar flux is

$$ Q = S_0 \frac{A_{cross-section}}{A_{surface}} $$

[ draw sketch of sphere and illuminated disk ]

where

• $A_{cross-section}$ = area of the illuminated disk = $\pi a^2$
• $A_{surface}$ = surface area of sphere = $4 \pi a^2$
• $a$ = radius of Earth

So flux in is $Q = S_0 / 4 = 341.3$ W m$^{-2}$

Flux out has two parts:

• Reflected solar radiation
• Emitted terrestrial (longwave) radiation

Introduce terminology / notation:

OLR = outgoing longwave radiation = terrestrial emissions to space

Define the planetary albedo:

• $\alpha$ = reflected solar flux / incoming solar flux
• Or reflected flux = $\alpha Q$ = 101.9 W m$^{-2}$ from data
• So from data, $\alpha \approx 0.3$

Define ASR = absorbed solar radiation \begin{align} ASR &= \text{ incoming flux – reflected flux} \\ &= Q - \alpha Q \\ &= (1-\alpha) Q \end{align}

Our energy budget then says

$$ \frac{dE}{dt} = (1-\alpha) Q - OLR $$

Note: This is a generically true statement. We have just defined some terms, and made the [very good] assumption that the only significant energy sources are radiative exchanges with space.

This equation is the starting point for EVERY CLIMATE MODEL.

But so far, we don’t actually have a MODEL. We just have a statement of a budget. To use this budget to make a model, we need to relate terms in the budget to state variables of the atmosphere-ocean system.

For now, the state variable we are most interested in is temperature – because it is directly connected to the physics of each term above.

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4. Using Python to compute emission to space

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Most of what follows is intended as a "fill in the blanks" exercise. We will practice writing some Python code while discussing the physical process of longwave emission to space.

Suppose the Earth behaves like a blackbody radiator with effective global mean emission temperature $T_e$.

Then

$$ OLR = \sigma T_e^4 $$

where OLR = "Outgoing Longwave Radiation", and $\sigma = 5.67 \times 10{-8}$ W m$^{-2}$ K$^{-4}$ the Stefan-Boltzmann constant

We can just take this as a definition of the emission temperature.

Looking back at the observations, the global, annual mean value for OLR is 238.5 W m$^{-2}$.

Calculate the emission temperature $T_e$

Rerranging the Stefan-Boltzmann law we get

$$ T_e = \left(\frac{\text{OLR}}{\sigma} \right)^{\frac{1}{4}} $$

First just use Python like a hand calculator to calculate $T_e$ iteractively:

Try typing a few different ways, with and without whitespace.

Python fact 1

extra spaces are ignored!

But typing numbers interactively is tedious and error prone. Let's define a variable called sigma

Python fact 2

We can define new variables interactively. Variables let us give names to things. Names make our code easy to understand.

Thoughts on emission temperature

What value did we find for the emission temperature $T_e$? How does it compare to the actual global mean surface temperature?

Is the blackbody radiator a good model for the Earth's emission to space?

A simple greenhouse model

The emission to space is lower because of the greenhouse effect, which we will study in detail later.

For now, just introduce a basic concept:

Only a fraction of the surface emission makes it out to space.

We will model the OLR as

$$ \text{OLR} = \tau \sigma T_s^4 $$

where $\tau$ is a number we will call the transmissivity of the atmosphere.

Let's fit this model to observations:

$$ \tau = \frac{\text{OLR}}{\sigma T_s^4} $$

Try calculating OLR for a warmer Earth at 292 K:

Naturally the emission to space is higher. By how much has it increased for this 4 degree warming?

Answer: 13.5 W m$^{-2}$. Okay but this is tedious and prone to error. What we really want to do is define a reusable function

Note a few things:

• The colon at the end of the first line indicates that there is more coming.
• The interpreter automatically indents the code for us (after the colon)
• The interpreter automatically colors certain key words
• We need to hit return one more time at the end to finish our function

Python fact 3

Indentations are not ignored! They serve to group together several lines of code.

we will see plenty of examples of this – in this case, the indentation lets the interpreter know that the code is all part of the function definition.

Python fact 4:

def is a keyword that defines a function.

Just like a mathematical function, a Python function takes one or more input arguments, performs some operations on those inputs, and gives back some resulting value.

Python fact 5:

return is a keyword that defines what value will be returned by the function.

Once a function is defined, we can call it interactively:

Python fact 6

The # symbol is used for comments in Python code.

The interpreter will ignore anything that follows # on a line of code.

Python fact 7:

print is a function that causes the value of an expression (or a list of expressions) to be printed to the screen.

(Don’t always need it, because by default the interpreter prints the output of the last statement to the screen, as we have seen).

Note also that we defined variables named sigma and epsilon inside our OLR function.

What happens if you try to print(epsilon)?

Python fact 8:

Variables defined in functions do not exist outside of that function.

Try declaring sigma = 2, then print(sigma). And try computing OLR(288) again. Did anything change?

Note that we didn’t really need to define those variables inside the function. We could have written the function in one line.

But sometimes using named variables makes our code much easier to read and understand!

Arrays with numpy

Now let’s try some array calculations:

• We have just created an array object.
• The linspace function creates an array of numbers evenly spaced between the start and end points.
• The third argument tells Python how many elements we want.

We will use the numpy package all the time. It is the basic workhorse of scientific computing with Python. We can't do much with arrays of numbers.

Does our OLR function work on an array of temperature values?

Now let’s assign these values to a new variable.

Now try again to compute OLR(288)

What do you get?

Python fact 9: Assigning a value to a named variable overwrites whatever was already assigned to that name.

Python is also case sensitive. If we had used olr to store the array, there would be no conflict.

Now let’s re-enter our function. Start typing def and then hit the “up arrow” key. What happens?

The editor gives us lots of useful keyboard shortcuts.

Here it’s looking up the last expression we entered that began with def. Saves a lot of time and typing!

Re-enter the function.

What happens if you use the up arrow without typing anything first?

Also, try typing history

This is very handy. The Python console is taking notes for you!

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Version information

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Credits

The author of this notebook is Brian E. J. Rose, University at Albany.

It was developed in support of ATM 623: Climate Modeling, a graduate-level course in the Department of Atmospheric and Envionmental Sciences

Development of these notes and the climlab software is partially supported by the National Science Foundation under award AGS-1455071 to Brian Rose. Any opinions, findings, conclusions or recommendations expressed here are mine and do not necessarily reflect the views of the National Science Foundation.

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