Learning from a simple model

A lot of what gets discussed here in relation to the greenhouse effect is relatively simple, and yet can be confusing to the lay reader. A useful way of demonstrating that simplicity is to use a stripped down mathematical model that is complex enough to include some interesting physics, but simple enough so that you can just write down the answer. This is the staple of most textbooks on the subject, but there are questions that arise in discussions here that don’t ever get addressed in most textbooks. Yet simple models can be useful there too.

I’ll try and cover a few ‘greenhouse’ issues that come up in multiple contexts in the climate debate. Why does ‘radiative forcing’ work as method for comparing different physical impacts on the climate, and why you can’t calculate climate sensitivity just by looking at the surface energy budget. There will be mathematics, but hopefully it won’t be too painful.

So how simple can you make a model that contains the basic greenhouse physics? Pretty simple actually. You need to account for the solar radiation coming in (including the impact of albedo), the longwave radiation coming from the surface (which depends on the temperature) and some absorption/radiation (the ’emissivity’) of longwave radiation in the atmosphere (the basic greenhouse effect). Optionally, you can increase the realism by adding feedbacks (allowing the absorption or albedo to depend on temperature), and other processes – like convection – that link the surface and atmosphere more closely than radiation does. You can skip directly to the bottom-line points if you don’t want to see the gory details.

The Greenhouse Effect

The basic case is set up like so: Solar radiation coming in is , where is the albedo, TSI the solar ‘constant’ and the factor 4 deals with the geometry (the ratio of the area of the disk to the area of the sphere). The surface emission is where is the Stefan-Boltzmann constant, and is the surface temperature and the atmospheric radiative flux is written , where is the emissivity – effectively the strength of the greenhouse effect. Note that this is just going to be a qualitative description and can’t be used to quantitatively estimate the real world values.

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