ATM 623: Climate Modeling¶
Brian E. J. Rose, University at Albany
Lecture 1: Planetary energy budget¶
About these notes:¶
This document uses the interactive IPython notebook format (now also called Jupyter). The notes can be accessed in several different ways:
- The interactive notebooks are hosted on
githubat 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 2015 (end of spring semester) are available on Brian's website.
Many of these notes make use of the climlab package, available at https://github.com/brian-rose/climlab
1. What is a Climate Model?¶
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.”
What is a Climate Model?
A minimal definition: 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.
Back to Wikipedia: http://en.wikipedia.org/wiki/Climate_model
Climate models use quantitative methods to simulate the interactions of the atmosphere, oceans, land surface, and ice. They are used for a variety of purposes from study of the dynamics of the climate system to projections of future climate. The most talked-about use of climate models in recent years has been to project temperature changes resulting from increases in atmospheric concentrations of greenhouse gases. All climate models take account of incoming energy from the sun as short wave electromagnetic radiation, chiefly visible and short-wave (near) infrared, as well as outgoing energy as long wave (far) infrared electromagnetic radiation from the earth. Any imbalance results in a change in temperature. Models can range from relatively simple to quite complex:
- A simple radiant heat transfer model that treats the earth as a single point and averages outgoing energy
- this can be expanded vertically (radiative-convective models), or horizontally
- finally, (coupled) atmosphere–ocean–sea ice global climate models discretise and solve the full equations for mass and >energy transfer and radiant exchange.
This is not a full list; for example "box models" can be written to treat flows across and within ocean basins. Furthermore, other types of modelling can be interlinked, such as land use, allowing researchers to predict the interaction between climate and ecosystems.
2. The observed global energy budget¶
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.
from IPython.display import Image
Image('../images/GlobalEnergyBudget.png')
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 $H_2O$ 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 these terms to vary under anthropogenic climate change?¶
3. Quantifying the planetary energy budget¶
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.
4. The simplest climate model:¶
The zero-dimensional energy balance model (EBM)¶
Suppose the Earth behaves like a blackbody radiator with effective global mean emission temperature $T_e$.
Then
$$ OLR = \sigma T_e^4 $$with $\sigma = 5.67 \times 10{-8}$ W m$^{-2}$ K$^{-4}$ the Stefan-Boltzmann constant
We can just take this as a definition of the emission temperature.
Also suppose that
$$ E = C T_s $$where $T_s$ is the global mean surface temperature, and $C$ is a constant – the effective heat capacity of the atmosphere- ocean column.
$C$ is in units of J m$^{-2}$ K$^{-1}$.
Why? We will look at this more carefully later, but essentially because the internal energy of a fluid is proportional to its temperature. Water and air heat up when you add energy to them!
We parameterize $E$ in terms of the surface temperature mostly because most of the heat capacity is in the oceans.
Now our budget is
$$C \frac{d T_s}{dt} = (1-\alpha) Q - \sigma T_e^4$$The climate system is at equilibrium when $ASR = OLR$ and the time derivative goes to zero.
This occurs when the emission temperature is precisely $$ \overline{T_e} = \bigg( \frac{(1-α)Q}{σ}\bigg)^{\frac{1}{4}} $$
with the overline denoting equilibrium.
Putting in numbers from our observed budget, we find $\overline{T_e} = 255$ K
- If the Earth emits as a blackbody at 255 K, the global energy budget is balanced and there will be no change in surface temperature
- If the Earth emits at a higher [lower] blackbody temperature, the atmosphere-ocean system will lose [gain] energy and the surface will cool [warm].
Notice that our time-dependent budget is a single equation in two unknowns, $T_s$ and $T_e$.
For now, we will assume that $C$, $\alpha$ and $Q$ are all fixed – though we will discuss reasons why all these terms vary throughout this course.
%install_ext http://raw.github.com/jrjohansson/version_information/master/version_information.py
%load_ext version_information
%version_information numpy, climlab
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, offered in Spring 2015.