The most common and most likely answer to this crossword clue is the 4 letter word GALE
I've seen this crossword clue in The Guardian
G
A
L
E
More clues leading to the result GALE
- Good beer gives you wind
- Good beer needed for powerful air force
- German booze giving you wind
- Old physician contracted a lot of wind
- Greek doctor doesn’t finish The Tempest
- Outburst from girlfriend over sweetheart
- Posh girl on ecstasy gets violent wind
- Winds thus will put ace golfer off course
- Good drink in pub giving you wind
- Strong draught beer with good head
Buddy explains!

The straight part
We know that "Tempest" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case tempest = gale.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does GALE mean?
GALE verb- To sing; charm; enchant.
- To cry; groan; croak.
- (of a person) To talk.
- (of a bird) To call.
- To sing; utter with musical modulations.
- A very strong wind, more than a breeze, less than a storm; number 7 through to 9 winds on the 12-step Beaufort scale.
- An outburst, especially of laughter.
- A light breeze.
- A song or story.
- To sail, or sail fast.
- A shrub, also called sweet gale or bog myrtle (Myrica gale), that grows on moors and fens.
- A periodic payment, such as is made of a rent or annuity.