# Greenhouse Model

## Contents

# 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#

## 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

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:

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

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 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.

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:

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

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

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.

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?