We are not seeing the sort of cycle My2cents means - we are in one of course, and on top of those are occasional short-term, minor spasms of cold or warm. However the main changes take place over tens or hundreds of thousands of years, not a mere couple of centuries.
We are in an Ice Age (a climatic oscillation between glacial and warm phases, colder and warmer than present); so either a warm period that can warm further, or its final thawing. If the latter, let us be thankful none of us will see its possible full effects.
However, the changes we are seeing now are too rapid to be entirely natural.
The fear of a "greenhouse" effect due to human activities enriching the carbon-dioxide proportion of the atmosphere, was predicted over 100 years ago but was largely ignored at the time for commercial and political reasons and perhaps the proposed timing of serious effects seeming long in human terms. It may also have fallen to a belief sincere then but which we now know wrong, that we can "conquer", "control" or "tame" Nature. We can certainly use and abuse Nature - but it's bigger than us and it bites back.
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I do not know if Mr. Trump was serious about Greenland, but if he was, he may have had other reasons to get his hands on Greenland. I don't know its geology but the country is losing its ice-cover, so eventually making any mineral reserves easier to find and exploit - which of course would be an equally valid reason for Denmark not to sell it.
Related to what Livvie wrote, he might also have considered siting military bases there; but I am not sure how hawkish he was.
If the Arctic opens sufficiently to regular shipping it would allow trade routes from NW Atlantic to North Pacific, along the tops of the American and Russian continents. A lot of Russian shipping does in fact follow the Siberian coast but usually behind ice-breakers. Examining a polar-plan map in my atlas, the Arctic Ocean is nearly encircled by land, and the pinch-point is the narrow Bering Strait. That is only about 50 miles wide at its narrowest, between the USA and Russia, with a border marked through the centre. Shipping-lanes might not be the major consideration though. Minerals on or under the sea-bed and on the many Arctic islands might be what really pre-occupies the governments.
There is another trade-route from Asia to Europe though, which the Russians and Chinese seem to be assiduously developing, and that is by rail. China has built some new lines across Tibet and Mongolia to the Trans-Siberian Railway, and it has not done that for nothing.
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Internally, though, the perceived threats in America may be from social stresses and instability due to parts of the country becoming no longer very habitable or suitable for agriculture; to more frequent weather extremes; and in some areas, to rising sea-level leading to both direct coastal and indirect riverine flooding. That could lead to many moving from State to State, abandoning unsaleable homes and farms in a re-run of the Dust Bowl years, but how would they be catered for or indeed welcomed, and how would the loss of farms and livelihoods in some states be compensated for in others?
These possible problems are of course by no means confined to America, but the geographical size and variety of the American continent and climate may mean a sort of collation of all outcomes in one nation.