Tutorial 6: Radiative Convective Equilibrium
Contents
Tutorial 6: Radiative Convective 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#
Building on the understanding of a one-dimensional radiative balance model from previous tutorial, in this tutorial students will learn about radiative-convective-equilibrium. 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 radiative-convective equilibrium model using the python package
climlab.Understand how this model builds off the one-dimensional radiative balance model used 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
from IPython.display import HTML
from matplotlib import animation
/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"
)
# 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
Plotting functions#
# @title Plotting functions
# make the videos at the end of the tutorial
plt.rcParams["animation.html"] = "jshtml"
# these three functions are used to make videos at the end of the tutorial
def initial_figure(model):
with plt.ioff(): # will hide the inital figure which will plot separate from the video otherwise
fig = plt.figure(figsize=(6, 6))
lines = []
skew = SkewT(fig, rotation=30)
# plot the observations
skew.plot(
Tglobal.level,
Tglobal,
color="black",
linestyle="-",
linewidth=2,
label="Observations",
)
lines.append(
skew.plot(
model.lev,
model.Tatm - climlab.constants.tempCtoK,
linestyle="-",
linewidth=2,
color="C0",
label="RC model (all gases)",
)[0]
)
skew.ax.legend()
skew.ax.set_ylim(1050, 10)
skew.ax.set_xlim(-60, 75)
# 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.ax.set_xlabel("Temperature ($^\circ$C)", fontsize=14)
skew.ax.set_ylabel("Pressure (hPa)", fontsize=14)
lines.append(
skew.plot(
1000,
model.Ts - climlab.constants.tempCtoK,
"o",
markersize=8,
color="C0",
)[0]
)
return fig, lines
def animate(day, model, lines):
lines[0].set_xdata(np.array(model.Tatm) - climlab.constants.tempCtoK)
lines[1].set_xdata(np.array(model.Ts) - climlab.constants.tempCtoK)
# lines[2].set_xdata(np.array(model.q)*1E3)
# lines[-1].set_text('Day {}'.format(int(model.time['days_elapsed'])))
# This is kind of a hack, but without it the initial frame doesn't appear
if day != 0:
model.step_forward()
return lines
# to setup the skewT and plot observations
def make_basic_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 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()
Video 1: Radiative-Convective Equilibrium#
# @title Video 1: Radiative-Convective Equilibrium
# Tech team will add code to format and display the video
Section 1: Reproducing Data from the Last Tutorial’s One-dimensional Radiative Equilibrium Model Using Climlab#
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
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
# take global, annual average
weight_factor = ds.gw / ds.gw.mean(dim="lat")
Qglobal = (ds.Q * weight_factor).mean(dim=("lat", "lon", "time"))
# 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)
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)
)
# 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"))
# 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()
skew = make_basic_skewT()
add_profile(skew, radmodel)
skew.ax.set_title("Pure Radiative Equilibrium", fontsize=18);
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[7], line 4
1 filename_sq = "cpl_1850_f19-Q-gw-only.cam.h0.nc"
2 url_sq = "https://osf.io/c6q4j/download/"
3 ds = xr.open_dataset(
----> 4 pooch_load(filelocation=url_sq, filename=filename_sq)
5 ) # ds = dataset
7 filename_ncep_air = "air.mon.1981-2010.ltm.nc"
8 url_ncep_air = "https://osf.io/w6cd5/download/"
Cell In[4], 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:
Section 2: Radiative-Convective Equilibrium#
From the plot you just made, one of the largest differences between observations and the pure radiation model with all gases lies in the lower atmosphere, where the surface air temperature is 20 degrees too warm and the 200 hPa pressure surface is 40 degrees too cold. What could be the issue?
One thing we have not included in our model yet is dynamics (motion of the air). The model’s temperature profile is what’s known as statically unstable (note this definition of stability is different than that used in the previous tutorials).
Here static means not due to wind, and unstable means the atmosphere wants to adjust to a different state because the surface air is relatively light and wants to rise into upper layers of the atmosphere. As the air rises, it creates convective turbulence (similar to boiling water, where convective circulation is introduced from heating water from below). The rising air, and the resultant turbulence, mixes the atmospheric column. This mixing often occurs across the troposphere, which is roughly the lowest 10km of the atmosphere. Most of the weather we experience lies in the troposphere.
When air rises adiabatically, it expands and cools due to the lower pressure. The rate of cooling depends on whether the air is saturated with water vapor. When rising air is unsaturated, it cools following the dry adiabats. If the air saturates, it cools at a lesser rate denoted by the moist adiabats (we did not have time to discuss these moisture effects in the mini-lecture).
To identify unstable atmospheric layers, let’s take another look at the SkewT plot, but this time we will plot the dry adiabats. We can then compare the rates of cooling of our model to these adiabats.
skew = make_skewT()
add_profile(skew, radmodel)
skew.ax.set_title("Pure Radiative Equilibrium", fontsize=18)
Near the surface, the reanalysis temperature profile is steeper than the dry adiabats. In these layers vertical motion is inhibited, and the surface conditions are considered stable. However, the model profile is shallower than the dry adiabats. In the model, the surface air is statically unstable, which would lead to convective mixing if this physical process was included in the model (the model currently only includes radiative processes). In this tutorial we will see whether including convective mixing in our model can bring the model closer to the reanalysis temperature profile.
To build a radiative-convective model we can take the radiative model we have already made and couple it to a convective model. Here the term couple implies there is communication between the models such that their effects are both incorporated into the final product, which in our case is the temperature profile.
# restate the model here for ease of coding
# make a model on same vertical domain as the water vapor data
mystate = climlab.column_state(lev=Qglobal.lev, water_depth=2.5)
# create the radiation model
rad = climlab.radiation.RRTMG(
name="Radiation (net)",
state=mystate,
specific_humidity=Qglobal.values,
timestep=climlab.constants.seconds_per_day,
albedo=0.25, # surface albedo, tuned to give reasonable ASR for reference cloud-free model
)
# create the convection model
conv = climlab.convection.ConvectiveAdjustment(
name="Convection",
state=mystate,
adj_lapse_rate=6.5, # the adiabatic lapse rate of the atmopshere
timestep=rad.timestep, # same timestep as radiation model
)
# couple the two components
rcm = climlab.couple([rad, conv], name="Radiative-Convective Model")
Now let’s run the radiation part to equilibrium, which should give us the same profile as in the previous section. Once we get this radiative equilibrium profile we will add convective mixing physics to the model and continue running it until it reaches radiative-convective equilibrium.
The new model does not actually resolve the actual vertical motion and mixing that occurs in convection. Instead, the model includes a parameterization for convection which automatically mixes regions of the atmospheric column that are statically unstable.
# run JUST the radiative component to equilibrium
for n in range(1000):
rcm.subprocess["Radiation (net)"].step_forward()
# compute diagnostics
rcm.compute_diagnostics()
# plot the resulting profile (our initial condition once we turn on the physics)
fig, lines = initial_figure(rcm)
# this animation can take a while
animation.FuncAnimation(fig, animate, 50, fargs=(rcm, lines))
Adding convective mixing to the initially unstable temperature profile leads to an instant mixing of air throughout the lower atmosphere, moving the profile towards the observations. The balance at play is between radiative process that warm the surface and cool the troposphere (lower atmosphere) as well as convection which moves heat away from the surface, leading to a colder surface and warmer troposphere. Note the differences in surface versus tropospheric temperatures in the new versus old equilibrium profiles.
Questions 2: Climate Connection#
The static instability was removed in the first time step! In reality, which process do you think changes temperature faster, convection or radiation?
What do you think the next step would be to move towards a more realistic climate model?
# to_remove explanation
"""
1. In nature, convection changes the temperature of the atmospheric column faster than radiative processes because convection involves quick movement of air (and heat) form one place to another. Our model takes this into account by having convective processes act instantaneously on unstablities due to unstable temperature profiles.
2. Next steps could include expanding in space (x,y,z) and including other processes (such as dynamics) and additional components of the climate system (ocean etc.). We'll talk about this more in Tutorial 7 where you will work with the Community Earth System Model (CESM).
""";
Coding Exercises 2#
Recreate the video above except using an isothermal atmosphere (uniform temperature profile) set to the surface temprature (rcm.state.Ts). Does the equilbrium profile look any different than the one you made before?
# set initial temperature profile to be the surface temperature (isothermal)
rcm.state.Tatm[:] = rcm.state.Ts
# compute diagnostics
_ = ...
# plot initial data
fig, lines = initial_figure(rcm)
# make animation - this animation can take a while
# animation.FuncAnimation(fig, animate, 50, fargs=(rcm, lines))
# to_remove solution
# set initial temperature profile to be the surface temperature (isothermal)
rcm.state.Tatm[:] = rcm.state.Ts
# compute diagnostics
_ = rcm.compute_diagnostics()
# plot initial data
fig, lines = initial_figure(rcm)
# make animation - this animation can take a while
animation.FuncAnimation(fig, animate, 50, fargs=(rcm, lines))
Summary#
In this tutorial, you explored the concept of radiative-convective equilibrium, learning to implement a Python model using the climlab package. You learned the limitations of a pure radiation model and the necessity to include atmospheric dynamics such as convection. Through coupling a radiation model with a convective model, you have simulated a more realistic atmospheric temperature profile. Even though the model does not directly resolve the vertical motion during convection, it incorporates a parameterization to automatically account for mixing in statically unstable regions. By the end of the tutorial, you could comprehend how this model expands on the previous energy balance models from tutorials 1-4.