Our eyes include lacrimal gland and lacrimal passage. The former is responsible for the produce of tears and the later for its elimination. In normal situation, some tears are evaporated into the airs and others inflow into the nasal cavity through lacrimal passage. Some people are sensitive in cold weather and stimulated things. So we often see the phenomenon that someone keep tearing in cold wind and can't control it.
Normally our eyes water when we are moved or sad in which the nerve system is stimulated. And the tears can be controlled. Your situation is a little weired. The cold wind must irritate your tear gland system by stimulating the eye nerves. You'd better go to see the doctor and have a check to see whether the eyes get infected.
Yes, it is supposed to be a natural phenomenon, because our eyes have to be moist all the time, and once the weather turned cold, our eyes will be dryer. Thus, our eyes will automatically produce some tears to keep them moist, that is the reason why you always shed tears in cold weather. But this is not a serious problem, if you really feel uncomfortable, just take some eye drops and drink more hot water. Perhaps things will turn out right. Good luck.