The Earth's magnetic field provides an important protection against the solar wind (for example, see Wikipedia on Earth's magnetic field and references therein). Mars may have lost its atmosphere because it did not retain such a magnetic field, although Venus does not have an intrinsic magnetic field either. This raises the question: what will happen to Earth's atmosphere if the global magnetic field would disappear? Would the solar wind blow it all away? If so, how long would it take until only a small part (<10 kPa sea level pressure) is left?
$\begingroup$ Mars still has an atmosphere. It may be thinner now than previously though. $\endgroup$– SivApr 22, 2014 at 18:26
5$\begingroup$ The surface escape velocity of Mars is also lower than Venus or Earth. $\endgroup$– winwaedApr 22, 2014 at 19:29
$\begingroup$ thats a good point winwaed, but wouldnt the solar forces be stronger? $\endgroup$– NeoApr 22, 2014 at 20:14
1$\begingroup$ “Size does matter.” $\endgroup$– gerrit ♦Apr 22, 2014 at 20:25
$\begingroup$ Difficult question: Measuring rates on Mars would be most comparable, but Mars probably also had 'atmospheric impact-erosion'. +The erosion probably happened faster, than changes in atmospheric screening (size of meteorite craters) can resolve. +Different magnitude of solar wind. I don't know of any numerical modelling, but that would probably give the best results for Earth. $\endgroup$– tobias47n9eApr 22, 2014 at 20:57
Atmospheric escape is a topic with a long research history. It is complex and is being addressed with both measurements and simulations.
For example, the question of atmospheric escape is still actively researched at Mars, and the MAVEN (Mars Atmosphere and Volatile Evolution) spacecraft mission is for example dedicated to this topic. Mars is a planet without a significant global magnetic field (although it does possess crustal magnetic anomalies); in fact its planetary dynamo was active and then stopped about 3.6 billion years ago, which considerably reduced the strength of its magnetic field.
So understanding the evolution of its atmosphere helps to relate to what would happen if Earth would loose its magnetic field. The question of atmospheric escape at Mars is relevant concerning the fate of water (geological evidences indicate the presence of liquid water in the past, which suggests a warm and dense atmosphere able to support it at that time).
Besides thermal atmospheric escape, there is also non thermal escape, including removal by asteroid impact (in the past) and by interaction between the atmosphere and the solar wind leading to pickup of atmospheric ions (in the past and continuing today).
Some examples of references to the scientific literature about atmospheric escape at terrestrial planets (abstracts):
Outgassing History and Escape of the Martian Atmosphere and Water Inventory
Nonthermal escape of the atmospheres of Venus, Earth, and Mars
Since atmospheric retention is largely dependent on escape velocity and temperature, removal of the Earth's magnetic field should not have a greatly noticeable effect, as current research shows that Earth's magnetic field changes the location of atmosphere loss due to the solar wind rather than eliminating it. Earth's temperature is not likely to change much without a magnetic field (with the current solar luminosity), so that too can be discounted.
It is far more likely that Earth's atmosphere will diminish as a result of increasing temperature as the sun expands due to an increase in helium fusion as it runs out of hydrogen, a phenomenon that would not be dependent on having or not having a magnetic field. That would be in roughly 5 to 7 billion years according to the Atmospheric Escape and the Formation and evolution of the Solar System Wikipedia articles.
If we discount solar expansion and make the convenient but incorrect assumption that solar wind and temperature will remain constant over time, then it is possible that Earth's atmosphere could be stripped in a timeframe on the order of many trillions of years. See the Atmospheric Escape Wikipedia article, and this How Vital Is a Planet's Magnetic Field Space.com article.
1$\begingroup$ Given that Venus does not have much of a magnetic field, has lower gravity and a higher temperature, and is closer to the sun where solar wind effects would be greater, and still has a very thick atmosphere presumably billions of years after its formation, I would say that solar wind effects would not be terribly significant at these distances on an object of sufficient size. $\endgroup$ Apr 22, 2014 at 23:22
$\begingroup$ That could be, I was specifically commenting on the suggestion that influx would equal outflux. The mass flux of the solar wind is insignificant. There is a net outflow of both hydrogen and oxygen from the Earth's atmosphere, fortunately there's a vast reservoir in the crust and oceans, so it's not a practical problem (for example, see Slapak (2013)). The question is, how much more would it be without a magnetosphere? $\endgroup$– gerrit ♦Apr 22, 2014 at 23:28
4$\begingroup$ Ok. Still, your answer remains unsourced, and it seems speculative, and therefore it is not very useful. Some references to peer-reviewed literature would be highly valuable. $\endgroup$– gerrit ♦Apr 22, 2014 at 23:34
1$\begingroup$ @Spießbürger, quite right. Please note my edited answer including references. It appears that my educated guesses weren't too far off, though. $\endgroup$ Apr 23, 2014 at 23:57
1$\begingroup$ @MontyWild But the question is about "how long will it take", not "when will it happen". $\endgroup$ Apr 24, 2014 at 5:18