9a[Morris, Edward E., 1898, Austral English: A Dictionary of Australasian Words, Phrases, and Usages, Macmillan, London]

...

Willy Willy, n. native name for a storm on North-west of Australia.

1894. 'The Age,' Jan. 20, p. 13, col. 4 [Letter by 'Bengalee']:

"Seeing in your issue of this morning a telegraphic report of a

'willy willy' in the north-west portion of West Australia, it

may be of interest to hear a little about these terrific storms

of wind and rain. The portion of the western coast most

severely visited by these scourges is said to be between the

North-wet Cape and Roebuck Bay; they sometimes reach as far

south as Carnarvon and north as far as Derby. The approach of

one of these storms is generally heralded by a day or too of

hot, oppressive weather, and a peculiar haze. Those having

barometers are warned of atmospheric disturbances; at other

times they come up very suddenly. The immense watercourses to

be seen in the north-west country, the bed of the Yule River,

near Roebourne, for instance, and many other large creeks and

rivers, prove the terrible force and volume of water that falls

during the continuance of one of these storms. The bed of the

Yule River is fully a mile wide, and the flood marks on some of

the trees are sufficient proof of the immense floods that

sometimes occur. Even in sheltered creeks and harbours the

wind is so violent that luggers and other small craft are blown

clean over the mangrove bushes and left high and dry, sometimes

a considerable distance inland. The willy willy is the name

given to these periodical storms by the natives in the

north-west."

1895. C. M. Officer, Private Letter:

"In the valley of the Murray between Swan Hill and Wentworth,

in the summer time during calm weather, there are to be seen

numerous whirlwinds, carrying up their columns of dust many

yards into the air. These are called by the name willy willy."

...

AB notes:

I found a reference to Austral English in The West Australian.

My references to willy-willy predate these by a decade or so.

Project Gutenberg has this: http://gutenberg.net.au/ebooks09/0900231h.html

9b[http://www.macquariedictionary.com.au]

...

2. Obsolete a cyclonic storm. Also, whirly-whirly; Especially Qld, whirly wind. [Aborig.; ? Yindjibarndi wili wili, or from Wembawemba wilang-wilang