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There is clear and ever increasing evidence of significant long-term changes in the earth's climate which will have a substantial impact on our lives and on the lives of the other living beings with whom we share this planet.  What is presented here is no more than a basic primer on climate change with brief accounts of Global Warming, the Greenhouse Effect and the Global Carbon Cycle.
For those who want to know more, the above links contain a mass of information on this vital but complex topic.
Perhaps the most spectacular example of the effects of global warming is the change that has occurred to the Portage Glacier in Alaska. A photograph of the glacier was taken in 1914 and another very recently from the same location. The glacier has effectively been replaced by a lake.

Global Warming

The following graph of global temperature anomaly (difference from the overall average temperature) for the period from 1850 to 2005 indicates, without a shadow of doubt, that global warming is happening.

 

The graph is provided by the Climate Research Unit of the University of East Anglia which is a good source of regularly updated data. The graph shows that there has been an increase in global temperature of nearly 0.8ºC over the last 100 years. The increases have been most pronounced in the winters of the northern hemisphere. The apparent discontinuity in the warming from the late 1940s to the mid 1970s was due to a cooling in the Northern hemisphere due to a mix of factors including variations in solar activity, unusually high concentrations of volcanic aerosols and increasing industrial pollutants coupled with successful efforts to reduce these. 
A good detailed account of this and other aspects of the problem can be found at Spencer Weart – The Discovery of Global Warming

The Greenhouse Effect

The solar radiation reaching the earth's atmosphere amounts to 343 Watts per sq m.. This is an average figure for the whole globe and the whole year. Of this, nearly 90 Watts are scattered and reflected back into space. A further 65 Watts of the longer wavelength radiation is absorbed by the atmosphere. This leaves nearly 190 Watts per sq m of the visible light and ultra-violet light which reaches the earth's surface. Of this 14 Watts are reflected (mostly from the oceans) leaving 175 Watts which are absorbed and heat the earth's surface. This heating results in the emission of infrared radiation back into the atmosphere. What happens is best described by looking at what happens in a greenhouse. Glass is almost totally  transparent to visible light which enters the greenhouse and is absorbed by the ground which heats up. The warmer the ground the more it emits infrared radiation. The glass, however is less transparent to infrared than to visible light thus the inside of the greenhouse gets warmer than the outside, and it goes on getting warmer and the intensity of the infrared radiation produced increases until the amount getting out  through the glass balances the incoming radiation. In the earth's atmosphere it is the presence of carbon dioxide, methane and some other gases that makes the atmosphere work like the glass in a greenhouse (hence the name greenhouse gases). The 175 Watts per sq m absorbed at the earth's surface warm it up.  It emits a infrared radiation a proportion of which is absorbed by the atmosphere. The warming of the earth's surface and the atmosphere continue until a balance is reached. Therefore, although it is counter-intuitive, most of the heating of the earth's atmosphere is through conduction and radiation from the earth's surface and not by the sun from above. It is this fact coupled with the variation in the amount of heating from the equator to the poles and the distribution of land and ocean that determines the major patterns of the earth's weather. It follows that any variation in the amount of heating of the atmosphere will have an effect on climate.
Over the long term the emission of radiation from the earth precisely balances the 343Watts per sq m of the incident radiation, while the temperatures of the atmosphere and the earth's surface are determined to a large extent by the levels of greenhouse gases, chiefly carbon dioxide and methane, in the atmosphere.

The Carbon Cycle

Due to the changes in temperature and day length associated with the changing seasons there is a clear annual cycle in the amount of carbon bound within living organisms. Carbon is taken from the atmosphere and incorporated in living organisms during the period of growth and increase in the spring and summer which is then released back into the atmosphere by death and decay during autumn and winter. The extent of the global exchange of carbon with the atmosphere amounts to nearly 170 billion tonnes of carbon every  year. Of this some 80 billion tonnes is connected with the land and 90 billion tonnes with the seas and oceans. Due to human activities of deforestation and the burning of fossil fuels an extra 6.5 billion tonnes of carbon is regularly being added to the atmosphere, chiefly as carbon dioxide but also some methane. Of this only about 3 billion tonnes is being taken up by the biosphere leaving a small but significant surplus of 3.5 billion tonnes which is accumulating year by year in the atmosphere. The concentration of Carbon dioxide in the atmosphere has been measured regularly at the observatory on the summit of Mauna Loa on Hawai'i.
 
The data show a steady increase in the levels of carbon dioxide over the last 40 years. Studies involving complex models of the earth's climate system have shown that the current warming of the earth can be attributed to the increase in the concentration of carbon dioxide and other greenhouse gases and that these increases can be attributed to human activity.