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Could a small meteor or natural nuclear explosion melt an ice cap on Earth to trigger a mini ice age without the event being observed by people?

Can the Earth have an overlaying thermal cycle that collects heat until it triggers long global winter?

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A small meteor/asteroid impact might trigger a nuclear winter, but it would be registered by people and it would be very unlikely to trigger an ice age unless the impact created a very large amount of dust in the atmosphere. The dust would need to stay in the atmosphere for a decade or more to trigger an ice age.

A natural nuclear explosion cannot be very large because as soon as critical mass is reached it will explode so the size of the explosion is in a way self limiting. If this happened under the world's major ice sheets it will melt a relatively large amount of ice but this would be registered by an increase in radioactivity in sea water (possibly many years after the explosion took place). So this would not trigger an ice age.

A large volcanic eruption can cool the earth for a few years but is unlikely to trigger an ice age unless it lasts for several decades, and yest again, this would be registered by people. You can take a look here to see what kind of magnitudes that have happened in our history https://www.ranker.com/list/the-worst-volcanic-eruptions-in-history/drake-bird

When you ask about an overlaying thermal cycle (as in heating until the temperature crashes), this is very unlikely to happen unless the temperature reaches a point where a very large part of life gets eradicated, as the temperature on earth goes up the ocean can hold less and less CO2 until at one point the oceans starts to release more CO2 than it absorbs from the atmosphere. This means the heating will not trigger cooling unless life on our planet is significantly reduced.

Water can hold less dissolved gasses when the temperature goes up, so at one point aerobic life in the ocean can no longer survive due to the lack of dissolved oxygen in the water.

This is just about how serious it might get if we keep on going like we are now.

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