At sea, surely eat with great self-discipline (9)

I believe the answer is:

austerely

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'at sea surely eat with great self-discipline' is the definition.
I can't judge whether this definition defines the answer.

'at sea surely eat' is the wordplay.
'at sea' indicates an anagram.
'surely'+'eat'='surelyeat'
'surelyeat' with letters rearranged gives 'AUSTERELY'.

Can you help me to learn more?

(Other definitions for austerely that I've seen before include "seriously" , "without extravagance" .)

I've seen this clue in The Times.

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