Physical Climatology Problem Set #4

A Global Energy Balance Model (EBM)

Background

The EBM is governed by the equation originally devised by both Sellers and Budyko in 1969:

(Shortwave in) = (Transport out) + (Longwave out)

which is formulated as

S(zone) [1 - Albedo(zone) ] = C x [ T(zone) - T(mean) ] + [ A + B T(zone) ]

where

S(zone) = the mean annual radiation incident at latitude (zone)

Albedo (zone) = the albedo at latitude (zone)

C = the transport coefficient (here set equal to 3.81 W m^-2^°C^-1)

T(zone) = the surface temperature at latitude (zone)

T(mean) = the mean global surface temperature

A and B are constants governing the longwave radiation loss which takes the linearized form here [see the Hartmann book Ch. 4, p. 114, Exercise 3; A = 204.0 W m^-2and B=2.17 W m^-2^°C^-1]

Note that if the surface temperature at latitude (zone) is less than -10°C the albedo is set to 0.68. The solar constant in the model is taken as 1370 W m^-2.

The EBM is designed to be used to examine the sensitivity of the predicted equilibrium climate to changes in the solar constant. If the default values for the variables A, B, C, and the albedo formulation are selected, an equilibrium climate which is quite close to the present-day situation is predicted for a fraction = 1 of the solar constant.

Once this equilibrium value of the solar constant has been seen, the user can modify the fraction of the solar constant prescribed and note the changes in the predicted climate. More importantly, the EBM permits the user to alter the albedo formulation, the latitudional transport and the parameters in the infrared radiation term and examine the sensitivity of the modified model.

How to run the model

The model is the fourth program called EBM in the program list at the Inquiry-Based Climate Models. The code can be run on any PC that has an internet. To run the model, you require Java[TM] plug-in for Applets. Click here to download the plug-in if the program does not run in your browser.

After you download the plug-in, simply click ebm.html will activate the program. Now you will see the following message on your screen.

A Global Energy Balance Model

There are various possibilities for changing the model climate. You can then test the sensitivity of this climate to changes in the solar constant. That you should observe the changes due to your changing of the model parameters is also of importance in understanding the nature of this model.

You will also see a main menu that states the following.

There are 3 main parameterization schemes within the model. You may make alterations to any or all of them at any one time. Any which you choose not to alter will be filled by default values.

The 3 parameterization schemes are: albedo & clouds, latitudinal transport, and longwave radiation to space.

There is a box named Run, which allows you to execute the program by clicking it.

In the parameterization of albedo & clouds, for instance, you will see the following by clicking albedo & clouds:

There are five things which you may alter

The temperature at which the surface becomes ice covered.

The albedo of this ice covered surface.

The albedo of the underlying ground.

Change the cloud amounts.

Change the cloud albedo.

In the parameterization of latitudinal transport, for instance, you will see the following by clicking latitudinal transport:

TRANSPORT

In this case you can alter the rate at which heat is transported around the model by varying the value of C in the following equation. Heat Flux = C x [ T(mean) - T(zone) ]. What is the value you want to use? 3.81

In the parameterization of longwave radiation to space, for instance, you will see the following by clicking longwave radiation to space:

LONGWAVE LOSS TO SPACE

The longwave loss to space is determined by the following equation. R = A + B x T(zone). Currently, A = 204.0; B = 2.17

When you click Run, you will be prompted by the following.

What fraction of the solar constant would you like? 0.5

where 0.5 is the default value and it should vary between 0 and 1.

Topics/problems for consideration

( i ). The traditional EBM simulations were examinations of the sensitivity of the model to changes in the solar constant. This will give you a 'feel' for the basic characteristics of the model and how it responds to a simple perturbation. Find the percentage change in solar constant to initiate a complete glaciation of the Earth.

[Hint: Using the default values of albedo, A, B, C, T_CRIT and the fractional cloud amount, determine what decrease in the solar constant (from a fraction of 1) is required just to glaciate the Earth completely (i.e., ice edge at 0°N).]

( ii ). (a) Various authors have suggested different values for the transport coefficient, C. For instant, Budyko (1969) originally used C=3.81 W m^-2^°C^-1 and Warren and Schneider (1979) used C = 3.74 W m^-2^°C^-1. How sensitive is the model's climate to the particular value of C?

( ii ). (b) Investigate the climate that results when using very small or very large values of C. How sensitive are these different climates to changes in the solar constant? Try to "predict" how you think the model will behave before you perform the experiment.

( iii ). (a) Observations show that land will be totally snow-covered during winter for an annual mean surface temperature of 0°C, and oceans totally ice-covered all year for a temperature of about -13°C. The model specifies a change from land/sea to snow/ice at -10°C. Alter this "critical" temperature and investigate the change in the climate and the climatic sensitivity to changing the solar constant.

( iii ). (b) The albedo over snow-covered areas can vary within the limits of 0.5-0.8 depending on vegetation type, cloud cover and snow/ice condition (e.g., ageing and impurities). Investigate the sensitivity of the simulated climate to changing the snow/ice albedo.

( iv ). (a) There have been many suggestions for the values of the constants A and B determining the longwave emission from the planet -- some have been dependent on cloud amount. Budyko (1969) originally used A = 202 W m^-2^°and B = 1.45 W m^-2^°C^-1. Cess (1976) suggested A = 212 W m^{-2 °} and B = 1.6 W m^-2^°C^-1. How do these different constants influence the climate and its sensitivity?

( iv ). (b) Holding A constant, just vary B and investigate the effect on the climate. What does a variation of B correspond to physically?

( v ). Repeat Exercise ( i ) with the values of A, B, C, and the albedo formulation which you believe are "best" (i.e. most physically realistic for the present-day climate). Once the Earth is just fully glaciated, begin to increase the fractional solar constant. Determine how much of an increase in the solar constant is required before the ice retreats from the equator. Do you understand the value?

Extra credits:

( vi ). How might the long timescale effects of the ocean be included in EBMs?

( vii ). How does the 'atmospheric circulation' encapsulated in an EBM compare with that simulated by a GCM?

( viii ). Can you find more than one way of modifying the model that produces the top-of-the-atmosphere fluxes prescribed (i.e. both the longwave out and the absorbed solar)? Check with your classmates - they may have a different solution (s) - can yours and theirs both be correct? In particular what do your experiments suggest about the opportunities for "validating" climate model results from restricted observations such as the satellite data?

( ix ). Return to the basic model and impose a different observed cloud distribution. You could try the observed mean cloud amounts for a given year (say, 2004) based on your best information. How do these values change surface climate?

( x ). Suppose CO₂ increases caused a decrease in cloud amount by between 10 and 20 percent. How is the climate affected? Could satellite measurements of zonal fluxes be a way of detecting such climate changes?