To answer your question, you change the issue - that will fire an event. Although I'm really not sure I understand the question at all.
However, the task is clear, and very easy. You'll need a plugin that provides "set custom field to current user" as a post-function on the "close" transition. There's one in the /Jira Suite Utilities that lets you set any custom field and provides a %current user% option for setting user-picker fields. I would (and do) use that one.
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