Greenhouse Model#

Learning objectives:#

  • Equilibrium states in the Earth system

  • Greenhouse effect

  • Develop a simple conceptual model

  • Integrate a model in time

  • How to set up sensitivity runs

After the exercise you should be able to answer the following questions:#

  • Why can we run climate simulations for many decades even though our predictability of weather events is very limited?

  • With this model we will perform sensitivity simulations that will show us important processes in the atmosphere.

Python Notebooks#

Greenhouse Model

Solution: Greenhouse Model

Energy balance of the one-layer model#

Assumptions#

Shortwave radiation:

  • Atmosphere is completely transparent to shortwave solar radiation

  • The surface absorbs shortwave radiation

Longwave radiation

  • Both surface and atmosphere emit radiation as blackbodies

  • Atmosphere is a single layer of air

  • Atmosphere is completely opaque to infrared radiation

  • Atmosphere radiates equally up and down

  • There are no other heat transfer mechanisms

../_images/one_layer_model.png

Fig. 5 One-layer greenhouse model. (Rose, Brian E. J. (2020, November 9). Insolation, The Climate Laboratory, https://brian-rose.github.io/ClimateLaboratoryBook/courseware/insolation.html).#

As with the simple energy balance model, we draw up the energy balances. This time we look at the energy balance of each layer individually and solve the two equations according to the equilibrium surface temperature.

The earth’s surface gains energy from the absorbed short-wave radiation and the long-wave radiation from the atmosphere. Only the outgoing long-wave radiation causes the Earth’s surface to lose energy. Thus, the energy balance equation can be written as follows:

(6)#\[ (1-\alpha) \cdot Q + \sigma T_a^4 = \sigma T_s^4. \]

The atmosphere, on the other hand, receives additional energy through the long-wave radiation of the Earth’s surface and loses energy in the form of long-wave radiation in the direction of the Earth’s surface and outer space. The budget can thus be written as

(7)#\[ \sigma T_s^4 = 2 \sigma T_a^4. \]

The factor 2 (on the right hand side) results from the fact that the atmosphere emits both towards the Earth’s surface and into space. Therefore, the surface temperature must have \(T_s>T_a\) with \(T_s = \sqrt[\leftroot{-2}\uproot{2}4]{2} \cdot T_a \approx 1.2 \cdot T_a\).

Task 1: Plug Eq. (7) into Eq. (6) and solve for the radiative equilibrium suface temperature \(T_e\). Since all the outgoing longwave radiation to outer space results from the atmosphere, the atmospheric temperture \(T_a\) is identical to the emission temperature \(T_e\).

Task 2: Where in the atmosphere do we find the calculated \(T_e\)? (Live-coding, {download}`Download netCDF-file<./files/air.mon.ltm.1981-2010.nc> Plot the mean temperature profile.

Task 3: What is the surface temperature with the single layer model? Why does the model overestimate the surface temperature?

Energy balance of the two-layer model#

In this part we want to introduce a more sophisticated two-layer model for the long-wave radiative transfer. In addition to an additional layer, we also assume that each layer absorbs only a part of the radiation. This conceptual model shows nicely how the greenhouse effect works.

Assumptions#

  • The atmosphere is transparent to shortwave radiation

  • Divide the atmosphere up into two layers of equal mass (the dividing line is thus at 500 hPa pressure level)

  • Each layer absorbs only a fraction \(\epsilon\) of whatever longwave radiation is incident upon it

  • Assume \(\epsilon\) is the same in each layer

This modelling approach is called the grey gas model. The designation grey is intended to indicate that there is no spectral dependence of absorption and emission.

../_images/two_layer_model.png

Fig. 6 Two-layer greenhouse model. (Rose, Brian E. J. (2020, November 9). Insolation, The Climate Laboratory, https://brian-rose.github.io/ClimateLaboratoryBook/courseware/insolation.html).#

In the following, we consider the upward beam of long-wave radiation, which is labelled U in fig. The earth’s surface emits long-wave energy according to the Stefan-Boltzmann law:

(8)#\[ U_0 = (\sigma T_s^4) \]

A fraction of this radiation is absorbed by layer 0. The remaining portion is passed on to layer 1 with the additional radiation coming from layer 0. The flux between layer 0 and layer 1 can then be written as

(9)#\[ U_1=(1-\epsilon)\sigma T_s^4+\epsilon \sigma T_0^4 \]

Similarly, the upward longwave radiation above layer 1 can be determined and written as

(10)#\[ U_2=(1-\epsilon)U_1+\epsilon \sigma T_1^4 \]

Task 4: Since there is no more atmosphere above layer 1, this upwelling beam is our OLR for this model. Plug \(U_1\) into the equation of \(U_2\) and solve the equation. What do the terms represent?

Task 5: Write a Python function for the OLR (Task 3).

Task 6: What happens if \(\epsilon\) is zero or one? What does this mean physically?

Task 7: We will tune our model so that it reproduces the observed global mean OLR given observed global mean temperatures. Determine the average temperatures (1000-500 hPa, 500 hPa to tropopause) for the two-layer model from the following sounding.

../_images/vertical_profile.png

Fig. 7 Atmospheric sounding from the NCEP-Reanalysis data. (Rose, Brian E. J. (2020, November 9). Insolation, The Climate Laboratory, https://brian-rose.github.io/ClimateLaboratoryBook/courseware/insolation.html).#

Task 8: Find graphically the best fit value of \(\epsilon\) using the observed temperatures and OLR.

Task 9: Write a Python function to calculate each term in the OLR. Plug-in the observed temperatures and the tuned value for epsilon and calculate the terms.

Task 10: Changing the level of emission by adding absorbers \(\epsilon=\epsilon+\Delta \epsilon\), e.g. by 10 %. 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? Which terms in the OLR go up and which go down?

Task 11: Calculate the radiative forcing for the previous simulation.

Task 12: What is the greenhouse effect for an isothermal atmosphere?

Task 13: For a more realistic example of radiative forcing due to an increase in greenhouse absorbers, we use our observed temperatures and the tuned value for epsilon. Assume an increase of epsilon by 2 %. What is the radiative forcing? What does this mean for the OLR?