ATM 623: Climate Modeling¶
Brian E. J. Rose, University at Albany
Lecture 17: Ice albedo feedback in the EBM¶
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
The annual mean EBM¶
the equation is
$$ C(\phi) \frac{\partial T_s}{\partial t} = (1-\alpha) ~ Q - \left( A + B~T_s \right) + \frac{D}{\cos\phi } \frac{\partial }{\partial \phi} \left( \cos\phi ~ \frac{\partial T_s}{\partial \phi} \right) $$Temperature-dependent ice line¶
Let the surface albedo be larger wherever the temperature is below some threshold $T_f$:
$$ \alpha\left(\phi, T(\phi) \right) = \left\{\begin{array}{ccc} \alpha_0 + \alpha_2 P_2(\sin\phi) & ~ & T(\phi) > T_f \\ a_i & ~ & T(\phi) \le T_f \\ \end{array} \right. $$%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import climlab
# for convenience, set up a dictionary with our reference parameters
param = {'A':210, 'B':2, 'a0':0.3, 'a2':0.078, 'ai':0.62, 'Tf':-10.}
model1 = climlab.EBM_annual( num_lat=180, D=0.55, **param )
print model1
Because we provided a parameter ai for the icy albedo, our model now contains several sub-processes contained within the process called albedo. Together these implement the step-function formula above.
The process called iceline simply looks for grid cells with temperature below $T_f$.
print model1.param
def ebm_plot( model, figsize=(8,12), show=True ):
'''This function makes a plot of the current state of the model,
including temperature, energy budget, and heat transport.'''
templimits = -30,35
radlimits = -340, 340
htlimits = -7,7
latlimits = -90,90
lat_ticks = np.arange(-90,90,30)
fig = plt.figure(figsize=figsize)
ax1 = fig.add_subplot(3,1,1)
ax1.plot(model.lat, model.Ts)
ax1.set_xlim(latlimits)
ax1.set_ylim(templimits)
ax1.set_ylabel('Temperature (deg C)')
ax1.set_xticks( lat_ticks )
ax1.grid()
ax2 = fig.add_subplot(3,1,2)
ax2.plot(model.lat, model.diagnostics['ASR'], 'k--', label='SW' )
ax2.plot(model.lat, -model.diagnostics['OLR'], 'r--', label='LW' )
ax2.plot(model.lat, model.diagnostics['net_radiation'], 'c-', label='net rad' )
ax2.plot(model.lat, model.heat_transport_convergence(), 'g--', label='dyn' )
ax2.plot(model.lat, model.diagnostics['net_radiation'].squeeze()
+ model.heat_transport_convergence(), 'b-', label='total' )
ax2.set_xlim(latlimits)
ax2.set_ylim(radlimits)
ax2.set_ylabel('Energy budget (W m$^{-2}$)')
ax2.set_xticks( lat_ticks )
ax2.grid()
ax2.legend()
ax3 = fig.add_subplot(3,1,3)
ax3.plot(model.lat_bounds, model.heat_transport() )
ax3.set_xlim(latlimits)
ax3.set_ylim(htlimits)
ax3.set_ylabel('Heat transport (PW)')
ax3.set_xlabel('Latitude')
ax3.set_xticks( lat_ticks )
ax3.grid()
return fig
model1.integrate_years(5)
f = ebm_plot(model1)
model1.diagnostics['icelat']
Add a small radiative forcing¶
The equivalent of doubling CO2 in this model is something like
$$ A \rightarrow A - \delta A $$where $\delta A = 4$ W m$^{-2}$.
deltaA = 4.
model2 = climlab.process_like(model1)
model2.subprocess['LW'].A = param['A'] - deltaA
model2.integrate_years(5, verbose=False)
plt.plot(model1.lat, model1.Ts)
plt.plot(model2.lat, model2.Ts)
The warming is polar-amplified: more warming at the poles than elsewhere.
Why?
Also, the current ice line is now:
model2.diagnostics['icelat']
There is no ice left!
Let's do some more greenhouse warming:
model3 = climlab.process_like(model1)
model3.subprocess['LW'].A = param['A'] - 2*deltaA
model3.integrate_years(5, verbose=False)
plt.plot(model1.lat, model1.Ts)
plt.plot(model2.lat, model2.Ts)
plt.plot(model3.lat, model3.Ts)
plt.xlim(-90, 90)
plt.grid()
In the ice-free regime, there is no polar-amplified warming. A uniform radiative forcing produces a uniform warming.
In-class investigation:¶
We will repeat the exercise from Lecture 14, but this time with albedo feedback included in our model.
- Solve the annual-mean EBM (integrate out to equilibrium) over a range of different diffusivity parameters.
- Make three plots:
- Global-mean temperature as a function of $D$
- Equator-to-pole temperature difference $\Delta T$ as a function of $D$
- Poleward heat transport across 35 degrees $\mathcal{H}_{max}$ as a function of $D$
- Choose a value of $D$ that gives a reasonable approximation to observations:
- $\Delta T \approx 45$ ºC
Use these parameter values:
param = {'A':210, 'B':2, 'a0':0.3, 'a2':0.078, 'ai':0.62, 'Tf':-10.}
print param
One possible way to do this:¶
Darray = np.arange(0., 2.05, 0.05)
model_list = []
Tmean_list = []
deltaT_list = []
Hmax_list = []
for D in Darray:
ebm = climlab.EBM_annual(num_lat=360, D=D, **param )
#ebm.subprocess['insolation'].s2 = -0.473
ebm.integrate_years(5., verbose=False)
Tmean = ebm.global_mean_temperature()
deltaT = np.max(ebm.Ts) - np.min(ebm.Ts)
HT = ebm.heat_transport()
#Hmax = np.max(np.abs(HT))
ind = np.where(ebm.lat_bounds==35.5)[0]
Hmax = HT[ind]
model_list.append(ebm)
Tmean_list.append(Tmean)
deltaT_list.append(deltaT)
Hmax_list.append(Hmax)
color1 = 'b'
color2 = 'r'
fig = plt.figure(figsize=(8,6))
ax1 = fig.add_subplot(111)
ax1.plot(Darray, deltaT_list, color=color1, label='$\Delta T$')
ax1.plot(Darray, Tmean_list, '--', color=color1, label='$\overline{T}$')
ax1.set_xlabel('D (W m$^{-2}$ K$^{-1}$)', fontsize=14)
ax1.set_xticks(np.arange(Darray[0], Darray[-1], 0.2))
ax1.set_ylabel('Temperature ($^\circ$C)', fontsize=14, color=color1)
for tl in ax1.get_yticklabels():
tl.set_color(color1)
ax1.legend(loc='center right')
ax2 = ax1.twinx()
ax2.plot(Darray, Hmax_list, color=color2)
ax2.set_ylabel('Poleward heat transport across 35.5$^\circ$ (PW)', fontsize=14, color=color2)
for tl in ax2.get_yticklabels():
tl.set_color(color2)
ax1.set_title('Effect of diffusivity on EBM with albedo feedback', fontsize=16)
ax1.grid()
4. Diffusive response to a point source of energy¶
Let's add a point heat source to the EBM and see what sets the spatial structure of the response.
We will add a heat source at about 45º latitude.
First, we will calculate the response in a model without albedo feedback.
param_noalb = {'A': 210, 'B': 2, 'D': 0.55, 'Tf': -10.0, 'a0': 0.3, 'a2': 0.078}
m1 = climlab.EBM_annual(num_lat=180, **param_noalb)
print m1
m1.integrate_years(5.)
m2 = climlab.process_like(m1)
point_source = climlab.process.energy_budget.ExternalEnergySource(state=m2.state)
ind = np.where(m2.lat == 45.5)
point_source.heating_rate['Ts'][ind] = 100.
m2.add_subprocess('point source', point_source)
print m2
m2.integrate_years(5.)
plt.plot(m2.lat, m2.Ts - m1.Ts)
plt.xlim(-90,90)
plt.grid()
The warming effects of our point source are felt at all latitudes but the effects decay away from the heat source.
Some analysis will show that the length scale of the warming is proportional to
$$ \sqrt{\frac{D}{B}} $$so increases with the diffusivity.
Now repeat this calculate with ice albedo feedback
m3 = climlab.EBM_annual(num_lat=180, **param)
m3.integrate_years(5.)
m4 = climlab.process_like(m3)
point_source = climlab.process.energy_budget.ExternalEnergySource(state=m4.state)
point_source.heating_rate['Ts'][ind] = 100.
m4.add_subprocess('point source', point_source)
m4.integrate_years(5.)
plt.plot(m4.lat, m4.Ts - m3.Ts)
plt.xlim(-90,90)
plt.grid()
Now the maximum warming does not coincide with the heat source at 45º!
Our heat source has led to melting of snow and ice, which induces an additional heat source in the high northern latitudes.
Heat transport communicates the external warming to the ice cap, and also commuicates the increased shortwave absorption due to ice melt globally!
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.
%install_ext http://raw.github.com/jrjohansson/version_information/master/version_information.py
%load_ext version_information
%version_information numpy, climlab