METEO 469
From Meteorology to Mitigation: Understanding Global Warming

One-Dimensional Energy Balance Model

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There are many ways one can generalize upon the zero-dimensional EBM. As we saw in the previous section, we can try to resolve the additional, vertical degree of freedom in the climate system through a very simple idealization—the one layer generalization of the zero-dimensional EBM. If for no other reason than the fact that the incoming solar radiation is symmetric with respect to longitude, but varies quite dramatically with latitude, the latitudinal degree of freedom is the next most important property to resolve if we wish to obtain further insights into the climate system using a still relatively simple and tractable model.

That brings us to the concept of the one-dimensional energy balance model, where we now explicitly divide the Earth up into latitudinal bands, though we treat the Earth as uniform with respect to longitude. By introducing latitude, we can now more realistically represent processes like ice feedbacks which have a strong latitudinal component, since ice tends to be restricted to higher latitude regions.

Recall that we had, for the linearized zero-dimensional gray body EBM, a simple balance:

C d T S dt = ( 1-α )S 4 -A-B T S
(1)
 
where α is the Earth's albedo and A and B are coefficients for the linearized representation of the 4th degree term.

Generalizing the zero-dimensional EBM, we can write a similar radiation and energy balance equation for each latitude band i:

C p d T i dt =( 1- α i ) S i -A-B T i
(2)
 
where i represents each latitude band.

We have now introduced some extremely important generalizations. The temperature T i , albedo α i , and incoming solar radiation T S are now functions of latitude, allowing us to represent the disparity in incoming shortwave radiation between equator and pole, and the strong potential latitudinal dependence of albedo with latitude—in particular, when the temperature T i for a particular latitude zone falls below freezing, we represent the increased accumulation of snow/ice in terms of a higher albedo. The global average temperature T S is computed by an appropriate averaging of the temperatures for the different latitude bands T i .

Recall that the disparity in received solar radiation between low and high latitudes leads to lateral heat transport over the surface of the Earth by the atmospheric circulation and ocean currents. In the absence of lateral transport, the poles will become increasingly cold and the equator increasingly warm. Clearly, we must somehow represent this meridional heat transport in the model if we expect realistic results. This can be done through a very crude representation of the process of heat advection through a term that is proportional to the difference between the temperature, F[ T i - T s ] where F is some appropriately chosen constant, and T S is the global average temperature. This term represents processes associated with lateral heat advection that tends to warm regions that are colder than the global average and cool regions that are warmer than the global average.

This gives the final form of our one-dimensional EBM:

C p d T i dt +F( T i - T s )=( 1- α i ) S i -A-B T i
(3)
 

The model is complex enough now that there is no way to simply write down the solution anymore. But we can solve the model mathematically, through a very simple and primitive form of something we will encounter much more of in the future—a numerical climate model.

Schematic of a one-dimensional Energy Balance Model.
Figure 5.2: Schematic of a one-dimensional Energy Balance Model.
Credit: NYU, Courant Institute of Mathematical Sciences (CIMS)
A Climate Modeling Primer, A. Henderson-Sellers and K. McGuffie, Wiley, pg. 58, (1987)

One of the most important problems that was first studied using this simple one-dimensional model was the problem of how the Earth goes into and comes out of Ice Ages. Use the links below to open the demonstration, which is in 3 parts.