Deeply; in a state of deep involvement or immersion (archaic form).
From Old English prefix 'a-' (in a state of) combined with 'deep,' similar to how 'asleep' and 'awake' work, though this archaic form survives mainly in poetic contexts.
Like 'adead,' this word shows how Old English used the prefix 'a-' to create state-of-being adverbs—a pattern that's mostly died out except for 'asleep,' 'awake,' and a few others.
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