Hi Cristine,
Strictly talking in terms of the tracking code dimension it will increment visits until it either: Expires (1 Week in this case) OR is overwritten by another Tracking Code OR tracking code cookie is deleted in browser somehow.
email (tracking code set and 1 week expiration starts)> direct (no tracking code) > natural search (no tracking code) = email clicks=1, and visits=3
email (tracking code set and 1 week expiration starts)> direct (no tracking code) > paid search (new tracking code set and 1 week expiration starts) = email clicks=1, email visits=2, paid search clicks=1, paid search visits=1
A good campaign tracking primer via Adam Greco:
http://blogs.adobe.com/digitalmarketing/analytics/campaign-tracking-inside-omniture-sitecatalyst/
Also main campaigns Help Link:
http://microsite.omniture.com/t2/help/en_US/sc/implement/campaign.html
Best,
Brian