Tutorial 5: Radiative Equilibrium
Contents
Tutorial 5: Radiative Equilibrium#
Week 1, Day 5, Climate Modeling
Content creators: Jenna Pearson
Content reviewers: Yunlong Xu, Will Gregory, Peter Ohue, Derick Temfack, Zahra Khodakaramimaghsoud, Peizhen Yang, Younkap Nina Duplex, Ohad Zivan, Chi Zhang
Content editors: Brodie Pearson, Abigail Bodner, Ohad Zivan, Chi Zhang
Production editors: Wesley Banfield, Jenna Pearson, Chi Zhang, Ohad Zivan
Our 2023 Sponsors: NASA TOPS and Google DeepMind
Tutorial Objectives#
In this tutorial students will run a one-dimensional radiative equilibrium model that predicts the global mean atmospheric temperature as a function of height. Much of the code shown here was taken from The Climate Laboratory by Brian Rose. Students are encouraged to visit this website for more tutorials and background on these models.
By the end of this tutorial students will be able to:
Implement a 1-D model that predicts atmospheric temperature as a function of height using the python package
climlab.Understand how this model builds off of the energy balance models developed in the previous tutorials.
Setup#
# note the conda install takes quite a while, but conda is REQUIRED to properly download the dependencies (that are not just python packages)
# !pip install condacolab &> /dev/null # need to use conda installation of climlab, pip won't work. condacolab is a workaround
# import condacolab
# condacolab.install()
# !mamba install -c anaconda cftime xarray numpy &> /dev/null # for decoding time variables when opening datasets
# !mamba install -c conda-forge metpy climlab &> /dev/null
# imports
import xarray as xr # used to manipulate data and open datasets
import numpy as np # used for algebra/arrays
import urllib.request # used to download data from the internet
import climlab # one of the models we are using
import matplotlib.pyplot as plt # used for plotting
import metpy # used to make Skew T Plots of temperature and pressure
from metpy.plots import SkewT # plotting function used widely in climate science
import pooch
import os
import tempfile
/home/wesley/miniconda3/envs/climatematch/lib/python3.10/site-packages/climlab/convection/akmaev_adjustment.py:142: NumbaDeprecationWarning: The 'nopython' keyword argument was not supplied to the 'numba.jit' decorator. The implicit default value for this argument is currently False, but it will be changed to True in Numba 0.59.0. See https://numba.readthedocs.io/en/stable/reference/deprecation.html#deprecation-of-object-mode-fall-back-behaviour-when-using-jit for details.
Akmaev_adjustment = jit(signature_or_function=Akmaev_adjustment)
Figure settings#
# @title Figure settings
import ipywidgets as widgets # interactive display
%config InlineBackend.figure_format = 'retina'
plt.style.use(
"https://raw.githubusercontent.com/ClimateMatchAcademy/course-content/main/cma.mplstyle"
)
Video 1: Radiative Equilibrium#
# @title Video 1: Radiative Equilibrium
# Tech team will add code to format and display the video
# helper functions
def pooch_load(filelocation=None, filename=None, processor=None):
shared_location = "/home/jovyan/shared/Data/tutorials/W1D4_Paleoclimate" # this is different for each day
user_temp_cache = tempfile.gettempdir()
if os.path.exists(os.path.join(shared_location, filename)):
file = os.path.join(shared_location, filename)
else:
file = pooch.retrieve(
filelocation,
known_hash=None,
fname=os.path.join(user_temp_cache, filename),
processor=processor,
)
return file
Section 1: Setting up the Radiative Equilibrium Model Using Climlab#
The energy balance model we used earlier today was zero-dimensional, yielding only the global mean surface temperature. We might ask, is it possible to construct a similar, one-dimensional, model for an atmospheric column to estimate the global mean temperature profile (i.e., including the height/\(z\) dimension). Additionally, can we explicitly include the effects of different gases in this model, rather than just parametrizing their collective effects through a single parameter \(\tau\)? The answer is yes, we can!
This model is too complex to construct from scratch, as we did in the previous tutorials. Instead, we will use a model already available within the python package climlab.
The model we will first use is a radiative equilbrium model. Radiative equilibrium models consider different layers of the atmosphere. Each of these layers absorbs and emits radiation depending on its constituent gases, allowing the model to calculate the radiation budget for each layer as radiative energy is transferred between atmospheric layers, the Earth’s surface, and space. Radiative equilibrium is reached when each layer gains energy at the same rate as it loses energy. In this tutorial you will analyze the temperature profile of this new model once it has reached equilibrium.
To set up this model, we will need information about some of the mean properties of the atmosphere. We are going to download water vapor data from the Community Earth System Model, a global climate model that we will go into detail on in the next tutorial, to use a variable called specific humidity. Specific humidity is the mass of water vapor per mass of a unit block of air. This is useful because water vapor is an important greenhouse gas.
filename_sq = "cpl_1850_f19-Q-gw-only.cam.h0.nc"
url_sq = "https://osf.io/c6q4j/download/"
ds = xr.open_dataset(
pooch_load(filelocation=url_sq, filename=filename_sq)
) # ds = dataset
ds
Downloading data from 'https://osf.io/c6q4j/download/' to file '/tmp/cpl_1850_f19-Q-gw-only.cam.h0.nc'.
---------------------------------------------------------------------------
KeyboardInterrupt Traceback (most recent call last)
Cell In[6], line 5
1 filename_sq = "cpl_1850_f19-Q-gw-only.cam.h0.nc"
2 url_sq = "https://osf.io/c6q4j/download/"
4 ds = xr.open_dataset(
----> 5 pooch_load(filelocation=url_sq, filename=filename_sq)
6 ) # ds = dataset
7 ds
Cell In[5], line 11, in pooch_load(filelocation, filename, processor)
9 file = os.path.join(shared_location, filename)
10 else:
---> 11 file = pooch.retrieve(
12 filelocation,
13 known_hash=None,
14 fname=os.path.join(user_temp_cache, filename),
15 processor=processor,
16 )
18 return file
File ~/miniconda3/envs/climatematch/lib/python3.10/site-packages/pooch/core.py:239, in retrieve(url, known_hash, fname, path, processor, downloader, progressbar)
236 if downloader is None:
237 downloader = choose_downloader(url, progressbar=progressbar)
--> 239 stream_download(url, full_path, known_hash, downloader, pooch=None)
241 if known_hash is None:
242 get_logger().info(
243 "SHA256 hash of downloaded file: %s\n"
244 "Use this value as the 'known_hash' argument of 'pooch.retrieve'"
(...)
247 file_hash(str(full_path)),
248 )
File ~/miniconda3/envs/climatematch/lib/python3.10/site-packages/pooch/core.py:803, in stream_download(url, fname, known_hash, downloader, pooch, retry_if_failed)
799 try:
800 # Stream the file to a temporary so that we can safely check its
801 # hash before overwriting the original.
802 with temporary_file(path=str(fname.parent)) as tmp:
--> 803 downloader(url, tmp, pooch)
804 hash_matches(tmp, known_hash, strict=True, source=str(fname.name))
805 shutil.move(tmp, str(fname))
File ~/miniconda3/envs/climatematch/lib/python3.10/site-packages/pooch/downloaders.py:226, in HTTPDownloader.__call__(self, url, output_file, pooch, check_only)
224 progress = self.progressbar
225 progress.total = total
--> 226 for chunk in content:
227 if chunk:
228 output_file.write(chunk)
File ~/miniconda3/envs/climatematch/lib/python3.10/site-packages/requests/models.py:816, in Response.iter_content.<locals>.generate()
814 if hasattr(self.raw, "stream"):
815 try:
--> 816 yield from self.raw.stream(chunk_size, decode_content=True)
817 except ProtocolError as e:
818 raise ChunkedEncodingError(e)
File ~/miniconda3/envs/climatematch/lib/python3.10/site-packages/urllib3/response.py:628, in HTTPResponse.stream(self, amt, decode_content)
626 else:
627 while not is_fp_closed(self._fp):
--> 628 data = self.read(amt=amt, decode_content=decode_content)
630 if data:
631 yield data
File ~/miniconda3/envs/climatematch/lib/python3.10/site-packages/urllib3/response.py:567, in HTTPResponse.read(self, amt, decode_content, cache_content)
564 fp_closed = getattr(self._fp, "closed", False)
566 with self._error_catcher():
--> 567 data = self._fp_read(amt) if not fp_closed else b""
568 if amt is None:
569 flush_decoder = True
File ~/miniconda3/envs/climatematch/lib/python3.10/site-packages/urllib3/response.py:533, in HTTPResponse._fp_read(self, amt)
530 return buffer.getvalue()
531 else:
532 # StringIO doesn't like amt=None
--> 533 return self._fp.read(amt) if amt is not None else self._fp.read()
File ~/miniconda3/envs/climatematch/lib/python3.10/http/client.py:466, in HTTPResponse.read(self, amt)
463 if self.length is not None and amt > self.length:
464 # clip the read to the "end of response"
465 amt = self.length
--> 466 s = self.fp.read(amt)
467 if not s and amt:
468 # Ideally, we would raise IncompleteRead if the content-length
469 # wasn't satisfied, but it might break compatibility.
470 self._close_conn()
File ~/miniconda3/envs/climatematch/lib/python3.10/socket.py:705, in SocketIO.readinto(self, b)
703 while True:
704 try:
--> 705 return self._sock.recv_into(b)
706 except timeout:
707 self._timeout_occurred = True
File ~/miniconda3/envs/climatematch/lib/python3.10/ssl.py:1274, in SSLSocket.recv_into(self, buffer, nbytes, flags)
1270 if flags != 0:
1271 raise ValueError(
1272 "non-zero flags not allowed in calls to recv_into() on %s" %
1273 self.__class__)
-> 1274 return self.read(nbytes, buffer)
1275 else:
1276 return super().recv_into(buffer, nbytes, flags)
File ~/miniconda3/envs/climatematch/lib/python3.10/ssl.py:1130, in SSLSocket.read(self, len, buffer)
1128 try:
1129 if buffer is not None:
-> 1130 return self._sslobj.read(len, buffer)
1131 else:
1132 return self._sslobj.read(len)
KeyboardInterrupt:
# the specific humidity is stored in a variable called Q
ds.Q
ds.time
however, we want an annual average profile:
# take global, annual average using a weighting (ds.gw) that is calculated based on the model grid - and is similar, but not identical, to a cosine(latitude) weighting
weight_factor = ds.gw / ds.gw.mean(dim="lat")
Qglobal = (ds.Q * weight_factor).mean(dim=("lat", "lon", "time"))
# print specific humidity profile
Qglobal
Now that we have a global mean water vapor profile, we can define a model that has the same vertical levels as this water vapor data.
# use 'lev=Qglobal.lev' to create an identical vertical grid to water vapor data
mystate = climlab.column_state(lev=Qglobal.lev, water_depth=2.5)
mystate
To model the absorption and emission of different gases within each atmospheric layer, we use the Rapid Radiative Transfer Model, which is contained within the RRTMG module. We must first initialize our model using the water vapor .
radmodel = climlab.radiation.RRTMG(
name="Radiation (all gases)", # give our model a name!
state=mystate, # give our model an initial condition!
specific_humidity=Qglobal.values, # tell the model how much water vapor there is
albedo=0.25, # this the SURFACE shortwave albedo
timestep=climlab.constants.seconds_per_day, # set the timestep to one day (measured in seconds)
)
radmodel
Let’s explore this initial state. Here Ts is the initial global mean surface temperature, and Tatm is the initial global mean air temperature profile.
radmodel.state
One of the perks of using this model is it’s ability to incorporate the radiative effects of individual greenhouse gases in different parts of the radiation spectrum, rather than using a bulk reduction in transmission of outgoing longwave radiation (as in our previous models).
Let’s display ‘absorber_vmr’, which contains the volume mixing ratio’s of each gas used in the radiative transfer model (these are pre-defined; and do not include the water vapor we used as a model input above). The volume mixing ratio describes the fraction of molecules in the air that are a given gas. For example, \(21\%\) of air is oxygen and so it’s volumn mixing ratio is 0.21.
radmodel.absorber_vmr
To look at carbon dioxide (CO2) in a more familiar unit, parts per million (by volume), we can convert and print the new value.
radmodel.absorber_vmr["CO2"] * 1e6
We can also look at all the available diagnostics of our model:
diag_ds = climlab.to_xarray(radmodel.diagnostics)
diag_ds
For example to look at OLR,
radmodel.OLR
Note. the OLR is currently 0 as we have not ran the model forward in time, so it has not calculated any radiation components.
Questions 1: Climate Connection#
Why do you think all gases, except ozone and water vapor, are represented by single values in the model?
# to_remove explanation
"""
1. The gases aside from ozone and water vapor are all assumed to be well mixed in the atmosphere. This means that we assume they have the same concentration anywhere you measure. However, we know from observations that ozone and water vapor are not well mixed in the atmosphere.
""";
Coding Exercises 1#
On the same graph, plot the annual mean specific humidity profile and ozone profiles.
fig, ax = plt.subplots()
# multiply Qglobal by 1000 to put in units of grams water vapor per kg of air
_ = ...
# multiply by 1E6 to get units of ppmv = parts per million by volume
_ = ...
# pressure decreases logarithmically with height in the atmosphere
# invert the axis so the largest value of pressure is lowest
ax.invert_yaxis()
# set y axis to a log scale
_ = ...
ax.set_ylabel("Pressure (hPa)")
ax.set_xlabel("Specific humidity (g/kg)")
# turn on the grid lines
_ = ...
# turn on legend
_ = ...
# to_remove solution
fig, ax = plt.subplots()
# multiply Qglobal by 1000 to put in units of grams water vapor per kg of air
_ = ax.plot(Qglobal * 1000.0, Qglobal.lev, label="Specific humidity (g/kg)")
# multiply by 1E6 to get units of ppmv = parts per million by volume
_ = ax.plot(radmodel.absorber_vmr["O3"] * 1e6, radmodel.lev, label="Ozone (ppmv)")
# pressure decreases logarithmically with height in the atmosphere
# invert the axis so the largest value of pressure is lowest
_ = ax.invert_yaxis()
# set y axis to a log scale
ax.set_yscale("log")
ax.set_ylabel("Pressure (hPa)")
ax.set_xlabel("Specific humidity (g/kg)")
# turn on the grid lines
_ = ax.grid()
# turn on legend
_ = ax.legend()
Section 2: Getting Data to Compare to the Model#
Before we run our model forward, we will download a reanalysis product from NCEP to get a sense of what the real global mean atmospheric temperature profile looks like. We will compare this profile to our model runs later.
filename_ncep_air = "air.mon.1981-2010.ltm.nc"
url_ncep_air = "https://osf.io/w6cd5/download/"
ncep_air = xr.open_dataset(
pooch_load(filelocation=url_ncep_air, filename=filename_ncep_air)
) # ds = dataset
# this is the long term monthly means (note only 12 time steps)
ncep_air.air
# need to take the average over space and time
# the grid cells are not the same size moving towards the poles, so we weight by the cosine of latitude to compensate for this
coslat = np.cos(np.deg2rad(ncep_air.lat))
weight = coslat / coslat.mean(dim="lat")
Tglobal = (ncep_air.air * weight).mean(dim=("lat", "lon", "time"))
Tglobal
Below we will define two helper funcitons to visualize the profiles output from our model with a SkewT plot. This is common way to plot atmospheric temperature in climate science, and the metpy package has a built in function to make this easier.
# to setup the skewT and plot observations
def make_skewT():
fig = plt.figure(figsize=(9, 9))
skew = SkewT(fig, rotation=30)
skew.plot(
Tglobal.level,
Tglobal,
color="black",
linestyle="-",
linewidth=2,
label="Observations",
)
skew.ax.set_ylim(1050, 10)
skew.ax.set_xlim(-90, 45)
# Add the relevant special lines
# skew.plot_dry_adiabats(linewidth=1.5, label = 'dry adiabats')
# skew.plot_moist_adiabats(linewidth=1.5, label = 'moist adiabats')
# skew.plot_mixing_lines()
skew.ax.legend()
skew.ax.set_xlabel("Temperature (degC)", fontsize=14)
skew.ax.set_ylabel("Pressure (hPa)", fontsize=14)
return skew
# to add a model derived profile to the skewT figure
def add_profile(skew, model, linestyle="-", color=None):
line = skew.plot(
model.lev,
model.Tatm - climlab.constants.tempCtoK,
label=model.name,
linewidth=2,
)[0]
skew.plot(
1000,
model.Ts - climlab.constants.tempCtoK,
"o",
markersize=8,
color=line.get_color(),
)
skew.ax.legend()
skew = make_skewT()
SkewT (also known as SkewT-logP) plots are generally used for much more complex reasons than we will use here. However, one of the benefits of this plot that we will utilize is the fact that pressure decreases approximately logarithmically with height. Thus, with a logP axis, we are showing information that is roughly linear in height, making the plots more intuitive.
Section 3: Running the Radiative Equilibrium Model Forward in Time#
We can run this model over many time steps, just like the simple greenhouse model, but now we can examine the behavior of the temperature profile rather than just the surface temperature.
There is no need to write out a function to step our model forward - climlab already has this feature. We will use this function to run our model to equilibrium (i.e., until OLR is balanced by ASR).
# take a single step forward to the diagnostics are updated and there is some energy imbalance
radmodel.step_forward()
# run the model to equilibrium (the difference between ASR and OLR is a very small number)
while np.abs(radmodel.ASR - radmodel.OLR) > 0.001:
radmodel.step_forward()
# check the energy budget to make sure we are really at equilibrium
radmodel.ASR - radmodel.OLR
Now let’s can compare this to observations.
skew = make_skewT()
add_profile(skew, radmodel)
skew.ax.set_title("Pure Radiative Equilibrium", fontsize=18);
Questions 3: Climate Connection#
The profile from our model does not match observations well. Can you think of one component we might be missing?
What effect do you think the individual gases play in determining this profile and why?
# to_remove explanation
"""
1. One thing we are currently lacking is physical processes (aside from radiation) in our model.
2. This is a hard question to answer! Luckily we can remove gases one at a time from our model to study their individual impact. Any ideas you have here can be tested with this model!
""";
Coding Exercises 3#
Create a second model called ‘Radiation (no H20)’ that lacks water vapor. Then re-create the plot above, but add on this extra profile without water vapor.
# make an exact clone of our existing model
radmodel_noH2O = climlab.process_like(radmodel)
# change the name of our new model
radmodel_noH2O.name = ...
# set the water vapor profile to all zeros
radmodel_noH2O.specific_humidity *= 0.0
# run the model to equilibrium
radmodel_noH2O.step_forward()
while np.abs(radmodel_noH2O.ASR - radmodel_noH2O.OLR) > 0.01:
radmodel_noH2O.step_forward()
# create skewT plot
skew = make_skewT()
# add profiles for both models to plot
for model in [...]:
...
# to_remove solution
# make an exact clone of our existing model
radmodel_noH2O = climlab.process_like(radmodel)
# change the name of our new model
radmodel_noH2O.name = "Radiation (no H2O)"
# set the water vapor profile to all zeros
radmodel_noH2O.specific_humidity *= 0.0
# run the model to equilibrium
radmodel_noH2O.step_forward()
while np.abs(radmodel_noH2O.ASR - radmodel_noH2O.OLR) > 0.01:
radmodel_noH2O.step_forward()
# create skewT plot
skew = make_skewT()
# add profiles for both models to plot
for model in [radmodel, radmodel_noH2O]:
add_profile(skew, model)
Summary#
In this tutorial, you’ve learned how to use the python package climlab to construct a one-dimensional radiative equilibrium model, and run it forward in time to predict the global mean atmospheric temperature profile. You’ve also visualized these results through SkewT plots.