Brian E. J. Rose, University at Albany
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.
This document uses the interactive Jupyter notebook format. The notes can be accessed in several different ways:
github at https://github.com/brian-rose/ClimateModeling_coursewareAlso 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
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# Ensure compatibility with Python 2 and 3
from __future__ import print_function, division
We can now use the concept of energy balance to ask what the temperature need to be in order to balance the energy budgets at the surface and the atmosphere, i.e. the radiative equilibrium temperatures.
The presence of the atmosphere above means there is an additional source term: downwelling infrared radiation from the atmosphere.
We call this the back radiation.
which means that $$ T_s = 2^\frac{1}{4} T_a \approx 1.2 T_a $$
So we have just determined that, in order to have a purely radiative equilibrium, we must have $T_s > T_a$.
The surface must be warmer than the atmosphere.
Now plug this into the surface equation to find
$$ \frac{1}{2} \sigma T_s^4 = (1-\alpha) Q $$and use the definition of the emission temperature $T_e$ to write
$$ (1-\alpha) Q = \sigma T_e^4 $$In fact, in this model, $T_e$ is identical to the atmospheric temperature $T_a$, since all the OLR originates from this layer.
Solve for the surface temperature: $$ T_s = 2^\frac{1}{4} T_e $$
Putting in observed numbers, $T_e = 255$ K gives a surface temperature of $$T_s = 303 ~\text{K}$$
This model is one small step closer to reality: surface is warmer than atmosphere, emissions to space generated in the atmosphere, atmosphere heated from below and helping to keep surface warm.
BUT our model now overpredicts the surface temperature by about 15ºC (or K).
Ideas about why?
Basically we just need to read our list of assumptions above and realize that none of them are very good approximations:
Let's generalize the above model just a little bit to build a slighly more realistic model of longwave radiative transfer.
We will address two shortcomings of our single-layer model:
Relaxing these two assumptions gives us what turns out to be a very useful prototype model for understanding how the greenhouse effect works.
This is called the grey gas model, where grey here means the emission and absorption have no spectral dependence.
We can think of this model informally as a "leaky greenhouse".
Note that the assumption that $\epsilon$ is the same in each layer is appropriate if the absorption is actually carried out by a gas that is well-mixed in the atmosphere.
Out of our two most important absorbers:
But we will ignore this aspect of reality for now.
In order to build our model, we need to introduce one additional piece of physics known as Kirchoff's Law:
$$ \text{absorptivity} = \text{emissivity} $$So if a layer of atmosphere at temperature $T$ absorbs a fraction $\epsilon$ of incident longwave radiation, it must emit
$$ \epsilon ~\sigma ~T^4 $$both up and down.
sympy packageThis two-layer grey gas model is simple enough that we can work out all the details algebraically. There are three temperatures to keep track of $(T_s, T_0, T_1)$, so we will have 3x3 matrix equations.
We all know how to work these things out with pencil and paper. But it can be tedious and error-prone.
Symbolic math software lets us use the computer to automate a lot of tedious algebra.
The sympy package is a powerful open-source symbolic math library that is well-integrated into the scientific Python ecosystem.
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import sympy
# Allow sympy to produce nice looking equations as output
sympy.init_printing()
# Define some symbols for mathematical quantities
# Assume all quantities are positive (which will help simplify some expressions)
epsilon, T_e, T_s, T_0, T_1, sigma = \
sympy.symbols('epsilon, T_e, T_s, T_0, T_1, sigma', positive=True)
# So far we have just defined some symbols, e.g.
T_s
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# We have hard-coded the assumption that the temperature is positive
sympy.ask(T_s>0)
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# Define these operations as sympy symbols
# And display as a column vector:
E_s = sigma*T_s**4
E_0 = epsilon*sigma*T_0**4
E_1 = epsilon*sigma*T_1**4
E = sympy.Matrix([E_s, E_0, E_1])
E
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# Define some new symbols for shortwave radiation
Q, alpha = sympy.symbols('Q, alpha', positive=True)
# Create a dictionary to hold our numerical values
tuned = {}
tuned[Q] = 341.3 # global mean insolation in W/m2
tuned[alpha] = 101.9/Q.subs(tuned) # observed planetary albedo
tuned[sigma] = 5.67E-8 # Stefan-Boltzmann constant in W/m2/K4
tuned
# Numerical value for emission temperature
#T_e.subs(tuned)
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U_0 = E_s
U_0
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Following this beam upward, we can write the upward flux from layer 0 to layer 1 as the sum of the transmitted component that originated below layer 0 and the new emissions from layer 0:
$$ U_1 = (1-\epsilon) U_0 + E_0 $$
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U_1 = (1-epsilon)*U_0 + E_0
U_1
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Continuing to follow the same beam, the upwelling flux above layer 1 is $$ U_2 = (1-\epsilon) U_1 + E_1 $$
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U_2 = (1-epsilon) * U_1 + E_1
Since there is no more atmosphere above layer 1, this upwelling flux is our Outgoing Longwave Radiation for this model:
$$ OLR = U_2 $$
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U_2
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The three terms in the above expression represent the contributions to the total OLR that originate from each of the three levels.
Let's code this up explicitly for future reference:
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# Define the contributions to OLR originating from each level
OLR_s = (1-epsilon)**2 *sigma*T_s**4
OLR_0 = epsilon*(1-epsilon)*sigma*T_0**4
OLR_1 = epsilon*sigma*T_1**4
OLR = OLR_s + OLR_0 + OLR_1
print( 'The expression for OLR is')
OLR
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fromspace = 0
D_2 = fromspace
Between layer 1 and layer 0 the beam contains emissions from layer 1:
$$ D_1 = (1-\epsilon)D_2 + E_1 = E_1 $$
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D_1 = (1-epsilon)*D_2 + E_1
D_1
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Finally between layer 0 and the surface the beam contains a transmitted component and the emissions from layer 0:
$$ D_0 = (1-\epsilon) D_1 + E_0 = \epsilon(1-\epsilon) \sigma T_1^4 + \epsilon \sigma T_0^4$$
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D_0 = (1-epsilon)*D_1 + E_0
D_0
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This $D_0$ is what we call the back radiation, i.e. the longwave radiation from the atmosphere to the surface.
In building our new model we have introduced exactly one parameter, the absorptivity $\epsilon$. We need to choose a value for $\epsilon$.
We will tune our model so that it reproduces the observed global mean OLR given observed global mean temperatures.
To get appropriate temperatures for $T_s, T_0, T_1$, let's revisit the global, annual mean lapse rate plot from NCEP Reanalysis data from the previous lecture.
First, we set $$T_s = 288 \text{ K} $$
From the lapse rate plot, an average temperature for the layer between 1000 and 500 hPa is
$$ T_0 = 275 \text{ K}$$Defining an average temperature for the layer between 500 and 0 hPa is more ambiguous because of the lapse rate reversal at the tropopause. We will choose
$$ T_1 = 230 \text{ K}$$From the graph, this is approximately the observed global mean temperature at 275 hPa or about 10 km.
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# add to our dictionary of values:
tuned[T_s] = 288.
tuned[T_0] = 275.
tuned[T_1] = 230.
tuned
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From the observed global energy budget we set
$$ OLR = 238.5 \text{ W m}^{-2} $$Subsitute in the numerical values we are interested in:
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# the .subs() method for a sympy symbol means
# substitute values in the expression using the supplied dictionary
# Here we use observed values of Ts, T0, T1
OLR2 = OLR.subs(tuned)
OLR2
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We have a quadratic equation for $\epsilon$.
Now use the sympy.solve function to solve the quadratic:
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# The sympy.solve method takes an expression equal to zero
# So in this case we subtract the tuned value of OLR from our expression
eps_solution = sympy.solve(OLR2 - 238.5, epsilon)
eps_solution
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There are two roots, but the second one is unphysical since we must have $0 < \epsilon < 1$.
Just for fun, here is a simple of example of filtering a list using powerful Python list comprehension syntax:
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# Give me only the roots that are between zero and 1!
list_result = [eps for eps in eps_solution if 0<eps<1]
print( list_result)
# The result is a list with a single element.
# We need to slice the list to get just the number:
eps_tuned = list_result[0]
print( eps_tuned)
We conclude that our tuned value is
$$ \epsilon = 0.586 $$This is the absorptivity that guarantees that our model reproduces the observed OLR given the observed tempertures.
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tuned[epsilon] = eps_tuned
tuned
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Even in this very simple greenhouse model, there is no single level at which the OLR is generated.
The three terms in our formula for OLR tell us the contributions from each level.
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OLRterms = sympy.Matrix([OLR_s, OLR_0, OLR_1])
OLRterms
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Now evaluate these expressions for our tuned temperature and absorptivity:
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OLRtuned = OLRterms.subs(tuned)
OLRtuned
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So we are getting about 67 W m$^{-2}$ from the surface, 79 W m$^{-2}$ from layer 0, and 93 W m$^{-2}$ from the top layer.
In terms of fractional contributions to the total OLR, we have (limiting the output to two decimal places):
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sympy.N(OLRtuned / 239., 2)
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Notice that the largest single contribution is coming from the top layer. This is in spite of the fact that the emissions from this layer are weak, because it is so cold.
Comparing to observations, the actual contribution to OLR from the surface is about 22 W m$^{-2}$ (or about 9% of the total), not 67 W m$^{-2}$. So we certainly don't have all the details worked out yet!
As we will see later, to really understand what sets that observed 22 W m$^{-2}$, we will need to start thinking about the spectral dependence of the longwave absorptivity.
Adding some extra greenhouse absorbers will mean that a greater fraction of incident longwave radiation is absorbed in each layer.
Thus $\epsilon$ must increase as we add greenhouse gases.
Suppose we have $\epsilon$ initially, and the absorptivity increases to $\epsilon_2 = \epsilon + \delta_\epsilon$.
Suppose further that this increase happens abruptly so that there is no time for the temperatures to respond to this change. We hold the temperatures fixed in the column and ask how the radiative fluxes change.
Do you expect the OLR to increase or decrease?
Let's use our two-layer leaky greenhouse model to investigate the answer.
The components of the OLR before the perturbation are
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OLRterms
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After the perturbation we have
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delta_epsilon = sympy.symbols('delta_epsilon')
OLRterms_pert = OLRterms.subs(epsilon, epsilon+delta_epsilon)
OLRterms_pert
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Let's take the difference
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deltaOLR = OLRterms_pert - OLRterms
deltaOLR
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To make things simpler, we will neglect the terms in $\delta_\epsilon^2$. This is perfectly reasonably because we are dealing with small perturbations where $\delta_\epsilon << \epsilon$.
Telling sympy to set the quadratic terms to zero gives us
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deltaOLR_linear = sympy.expand(deltaOLR).subs(delta_epsilon**2, 0)
deltaOLR_linear
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Recall that the three terms are the contributions to the OLR from the three different levels. In this case, the changes in those contributions after adding more absorbers.
Now let's divide through by $\delta_\epsilon$ to get the normalized change in OLR per unit change in absorptivity:
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deltaOLR_per_deltaepsilon = \
sympy.simplify(deltaOLR_linear / delta_epsilon)
deltaOLR_per_deltaepsilon
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Now look at the sign of each term. Recall that $0 < \epsilon < 1$. Which terms in the OLR go up and which go down?
THIS IS VERY IMPORTANT, SO STOP AND THINK ABOUT IT.
The contribution from the surface must decrease, while the contribution from the top layer must increase.
When we add absorbers, the average level of emission goes up!
We just worked out that whenever we add some extra absorbers, the emissions to space (on average) will originate from higher levels in the atmosphere.
What does this mean for OLR? Will it increase or decrease?
To get the answer, we just have to sum up the three contributions we wrote above:
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R = -sum(deltaOLR_per_deltaepsilon)
R
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Is this a positive or negative number? The key point is this:
It depends on the temperatures, i.e. on the lapse rate.
Stop and think about this question:
If the surface and atmosphere are all at the same temperature, does the OLR go up or down when $\epsilon$ increases (i.e. we add more absorbers)?
Understanding this question is key to understanding how the greenhouse effect works.
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R.subs([(T_0, T_s), (T_1, T_s)])
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which then simplifies to
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sympy.simplify(R.subs([(T_0, T_s), (T_1, T_s)]))
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For an isothermal atmosphere, there is no change in OLR when we add extra greenhouse absorbers. Hence, no radiative forcing and no greenhouse effect.
Why?
The level of emission still must go up. But since the temperature at the upper level is the same as everywhere else, the emissions are exactly the same.
For a more realistic example of radiative forcing due to an increase in greenhouse absorbers, we can substitute in our tuned values for temperature and $\epsilon$.
We'll express the answer in W m$^{-2}$ for a 1% increase in $\epsilon$.
The three components of the OLR change are
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deltaOLR_per_deltaepsilon.subs(tuned) * 0.01
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And the net radiative forcing is
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R.subs(tuned) * 0.01
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So in our example, the OLR decreases by 2.2 W m$^{-2}$, or equivalently, the radiative forcing is +2.2 W m$^{-2}$.
What we have just calculated is this:
Given the observed lapse rates, a small increase in absorbers will cause a small decrease in OLR.
The greenhouse effect thus gets stronger, and energy will begin to accumulate in the system -- which will eventually cause temperatures to increase as the system adjusts to a new equilibrium.
In the previous section we:
A key question in climate dynamics is therefore this:
What sets the lapse rate?
It turns out that lots of different physical processes contribute to setting the lapse rate.
Understanding how these processes acts together and how they change as the climate changes is one of the key reasons for which we need more complex climate models.
For now, we will use our prototype greenhouse model to do the most basic lapse rate calculation: the radiative equilibrium temperature.
We assume that
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# Upwelling and downwelling beams as matrices
U = sympy.Matrix([U_0, U_1, U_2])
D = sympy.Matrix([D_0, D_1, D_2])
# Net flux, positive up
F = U-D
F
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# define a vector of absorbed radiation -- same size as emissions
A = E.copy()
# absorbed radiation at surface
A[0] = F[0]
# Compute the convergence
for n in range(2):
A[n+1] = -(F[n+1]-F[n])
A
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radeq = sympy.Equality(A, sympy.Matrix([(1-alpha)*Q, 0, 0]))
radeq
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Just as we did for the 1-layer model, it is helpful to rewrite this system using the definition of the emission temperture $T_e$
$$ (1-\alpha) Q = \sigma T_e^4 $$
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radeq2 = radeq.subs([((1-alpha)*Q, sigma*T_e**4)])
radeq2
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In this form we can see that we actually have a linear system of equations for a set of variables $T_s^4, T_0^4, T_1^4$.
We can solve this matrix problem to get these as functions of $T_e^4$.
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# Solve for radiative equilibrium
fourthpower = sympy.solve(radeq2, [T_s**4, T_1**4, T_0**4])
fourthpower
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This produces a dictionary of solutions for the fourth power of the temperatures!
A little manipulation gets us the solutions for temperatures that we want:
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# need the symbolic fourth root operation
from sympy.simplify.simplify import nthroot
fourthpower_list = [fourthpower[key] for key in [T_s**4, T_0**4, T_1**4]]
solution = sympy.Matrix([nthroot(item,4) for item in fourthpower_list])
# Display result as matrix equation!
T = sympy.Matrix([T_s, T_0, T_1])
sympy.Equality(T, solution)
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In more familiar notation, the radiative equilibrium solution is thus
\begin{align} T_s &= T_e \left( \frac{2+\epsilon}{2-\epsilon} \right)^{1/4} \\ T_0 &= T_e \left( \frac{1+\epsilon}{2-\epsilon} \right)^{1/4} \\ T_1 &= T_e \left( \frac{ 1}{2 - \epsilon} \right)^{1/4} \end{align}Plugging in the tuned value $\epsilon = 0.586$ gives
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Tsolution = solution.subs(tuned)
# Display result as matrix equation!
sympy.Equality(T, Tsolution)
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Now we just need to know the Earth's emission temperature $T_e$!
(Which we already know is about 255 K)
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# Here's how to calculate T_e from the observed values
sympy.solve(((1-alpha)*Q - sigma*T_e**4).subs(tuned), T_e)
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# Need to unpack the list
Te_value = sympy.solve(((1-alpha)*Q - sigma*T_e**4).subs(tuned), T_e)[0]
Te_value
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# Output 4 significant digits
Trad = sympy.N(Tsolution.subs([(T_e, Te_value)]), 4)
sympy.Equality(T, Trad)
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Compare these to the values we derived from the observed lapse rates:
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sympy.Equality(T, T.subs(tuned))
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The radiative equilibrium solution is substantially warmer at the surface and colder in the lower troposphere than reality.
This is a very general feature of radiative equilibrium, and we will see it again very soon in this course.
Did we need sympy to work all this out? No, of course not. We could have solved the 3x3 matrix problems by hand. But computer algebra can be very useful and save you a lot of time and error, so it's good to invest some effort into learning how to use it.
Hopefully these notes provide a useful starting point.
You are now ready to tackle Assignment 5, where you are asked to extend this grey-gas analysis to many layers.
For more than a few layers, the analytical approach we used here is no longer very useful. You will code up a numerical solution to calculate OLR given temperatures and absorptivity, and look at how the lapse rate determines radiative forcing for a given increase in absorptivity.
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%load_ext version_information
%version_information sympy
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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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