These ideas have been seriously considered by scientists but may look over ambitious.
Increasing Plankton
Where in areas of the sea too warm for plankton to survive, colder water rich in nutrients from the depths are churned up, plankton and life become abundant. The growth of plankton absorbs CO2 from the sea, trapping the carbon as it dies and falls to the ocean floor. The potential in southern oceans would be to remove 1 billion tonnes of carbon a year (out of 7 billion tonnes a year man made. This would reduce the balance from 4 billion to 3 billion put into the atmosphere - see global page.)
Plankton growth emits dimethyl sulphide gas which creates cloud formation which in turn reflect the suns rays, cooling the earth.
Pumping nutrient rich water from 200m depth (650 feet) to decrease CO2 in atmosphere.
An idea to pump up water from 200m depth has been suggested by Professor Lovelock and Chris Rapley by installing 200m (650 foot) pipes 10m in diameter, buoyed up at one end, with a non return valve at the other, which with wave action would pump up say 5 tonnes of water a second.
Proposed locations of these hundreds of thousands of giant pipes are Gulf of Mexico, Christmas Island(NE australia), the great barrier reef, Atoll Islands (Indian Ocean).
Powdered Iron to be poured into Pacific to encourage Plankton growth and decrease CO2 in atmosphere.
Planktos hopes to increase plankton in the Pacific by 60 million tons so as to absorb CO2 and in 2 years take 3-5 million tons (7ppm) of carbon into the sea bed to be locked up.
Breeding Algae with high CO2 absorption rate
By selecting out algae with genomes linked to high CO2 absorption, colonies could be developed to increase the rate of CO2 draw down.
Synthetic trees to extract CO2 from the air
100 foot synthetic trees, using sodium hydroxide to convert CO2 to Sodium Carbonate, which would be stored deep underground. One tree would extract 90,000 tonnes of CO2 per year.
Giant sunshade in space to reflect the sun
To divert 2% of suns rays from earth a 100,000 mile wide shade, 1 million miles from earth, sitting in zero gravity consisting of 16 trillion 2 foot dia. glass mirrors 3 microns thick has been suggested. The total weight would be 20 million tons. the cost should be a few trillion dollars less than 0.5% of world GDP.
Cloud making yachts in southern ocean to reflect the sun
The idea is to atomize sea water and spray the vapor into the atmosphere which would reflect light. 1000 vessels putting 5 tonnes water a second would reflect 3% of light. The vessels would be powered by vertical cylinders that rotate with the wind. They would be positioned off Greenland in summer and repositioned in the southern ocean in the southern summer.
Cloud formation, cosmic rays and CERN
Cosmic rays are believed to encourage cloud formation which could reflect sunlight. One of the many possible benefits of the Large Hadron Collider experiments at CERN might be to research this.
Sulphur dioxide (SO2) screen to reflect the sun.
In 1991 Mt Pinotubo, in SE Asia erupted and put a million tonnes of SO2 into the atmosphere at high altitude for 2 years. The average world temperature dropped by 0.6 deg C average. The idea suggested is to put sulphur into the stratosphere by fitting sulphur dispensers into the worlds airliners or by using rockets. The disadvantage is acid rain and damage to the ozone layer.
Comment
One would expect that expense and feasibility need more detail assessment and it does not address the problem of oil etc running out. The more sensible approach would seem to be to reduce CO2 by green methods of generation and car propulsion, including nuclear. More detail on Professor Lovelocks blog.
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