Folklore in the news means cold on the way

The early arrival of eight Bewick's Swans (Cygnus columbianus) in Gloucestershire has, surprisingly, made the national press. The reports aren't about how attractive these small swans are. Rather, they focus on the distances they travel. 'Prone to ceaseless wanderlust', in the words of one ornithologist, Bewick's Swans fly some 2,500 miles from Russia/Siberia to overwinter in Britain.
More particularly, the press are interested in the date of their arrival. These birds have arrived a couple of weeks earlier than last year's migrants, and journalists have started reaching for Folklore. This, fairly standard version, was in the Telegraph: 'According to folklore, their early arrival signals the start of a long, harsh winter'. This makes a sort of sense, as Bewick's Swans fly south-ish to get away from Arctic winters, and early movement might indicate an early worsening of the weather behind them.
Looking at Richard Inwards's 1893 collection Weather Lore: A Collection of Proverbs, Sayings & Rules Concerning the Weather (repr. London: Senate, 1994), turns up a definite association of swans with bad weather. Inwards notes a Scottish belief that 'When the white swan visits the Orkneys, expect a continued severe winter' (p. 134). More generally, their flight is associated with rain or hurricanes. The rain connection seems quite venerable: Inwards quotes Dryden's translation of Vergil:

The swans that sail along the silvery flood,

And dive with stretching necks to search their food,
Then lave their backs with sprinkling dews in vain,
And stem the stream to meet the promised rain

They're a lovely sight, but it might be time to wrap up warm.