Because cold air is dry air and or lungs can’t work well unless the air is humid enough.
First of all, humidity is measured as how much water vapor is mixed in the ambient air. A 100% humidity means the air can’t take any more water, so a small pool won’t evaporate and some water will likely condensate in all surfaces trying to run from the air. The maximum quantity the air can take water depends on temperature. The higher the temperature, the more water vapor the air can take.
When we breath, air goes inside our lungs.
As our temperatures are constant the air in contact with our respiratory ways will most likely be heated or chilled by our bodies, so the humidity will be different.
If the air humidity is too low, it would dry our lungs and it is not a good thing! A dry lung can’t work! So we have organs in our nose that humidify the air entering or lungs. They work by giving out water in our noses, so some of it will evaporate.
So if you are in a cold environment, the air can’t take too much water. When you breath, the air is heated by your body, so it becomes dry, as it can suddenly take much more water vapor. As dry air is bad for your lungs, your nose try to humidify it by expelling water. And that’s why your nose start to run. It is trying to give out more water to the air before it reaches your lungs.
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