There’s this particular interjection-like usage of there that I think belongs to the spoken register and maybe doesn’t refer to anything; as in:
- Hi there!
- You alright there?
Why is it used, and what meaning (if any) does it serve to convey?
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Hi there! is a greeting. There refers to the position that the other person is in, so it is an adverb. It can also serve to attract attention.
Hi over there! or Hello over there works the same way as Hi there or Hello there, except that the distance between the two people is greater. The two people could be across a room or across the globe.
I think you you are right about social register, as Hi there! seems to me to be informal or at least not formal. This is also indicated by Hi.
The there in You all right there is also an adverb. You can also say things such as Hey there! and You there! and not say it in a friendly way. You can also say Who goes there?
Another stock phrase is
There she goes!
where, again, there is an adverb and is used to attract attention to whatever she is, be she a female person, a boat, or, in this case just about anything.