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They both mean rain with a thunder?

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No, there is a little difference between them.

嵐 means storm, tempest (without rain)

雷雨 means thunderstorm (雷 + 雨 = thunder + rain)

Little note:

嵐(あらし)is also Japanese boy band.

References:

https://jisho.org/search/%20%E5%B5%90 https://jisho.org/search/%20%E9%9B%B7%E9%9B%A8

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