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In the sentence 'I am happy that you are here', 'that you are here' acts as a subordinate clause.

However, I am unsure what type of subordinate clause it is: i.e. I'm not sure if it's an adverbial, adjectival, or noun clause.

Anyone able to help?

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This is an adverbial clause as adverbial clauses are used among others for C³--- cause, condition and contrast.

I am happy that (because) you are here.

The adjective happy is modified by the subordinate adverbial clause. Adverbs mostly modify verbs, adjectives or another adverb in a sentence.

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