From a definition of 降り続く：

> 雨や雪などが、何日も続けて降る。「—・く長雨にうんざりする」

If you boil this down to 雨が続けて降る it appears to make no sense if you consider
続ける as a transitive verb. What's going on?
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question
transitivity
Can 続けて be used intransitively?
I believe it is related to the construction

Stem+続ける which can be attached even to intransitive verbs.

I guess maybe it's just an inverted version of 降り続ける.
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2022-01-22T20:06:57.253
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answer
This 続けて is "continuously" that modifies 降る.

This is a te-form used _like an adverb_ to modify how the action described by
the second verb is done. 続けて can be remembered almost as a distinct adverb,
but you can read the following questions for grammatical explanations:

  * [How can verb て become an adverb?](https://japanese.stackexchange.com/q/34647/5010)
  * [立ってそばを食べる vs 立ちながらそばを食べる](https://japanese.stackexchange.com/q/62592/5010)
  * [て form and adverbial meaning](https://japanese.stackexchange.com/q/38769/5010)
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answer