How to alert users when new posts appear under search query?

I have been hunting the web for days for this feature, and I am kind of surprised it's so hard to find / nobody really talks about it!

I have a Wordpress site with a custom post type that can be pulled by different search attributes. I am looking for a way for users to subscribe to a specific search query string, and once per week, get notified automatically via email when new posts that match their search query string appear.

For example when somebody searches for www.mysite.com/?Location=CaliforniaRadius=25, a list of results come up, and if that user subscribes to this search result, then once per week he/she will be notified of new posts matching that search query

I am open to paid plugins as well, it's just I haven't been able to find one that can do this.

I am comfortable with capturing additional information from users when they're signing up, using jQuery and populating hidden fields' values, etc., if this would help with the end goal.

Edit: I thought of an idea of users subscribing to the RSS feed for each search query, since it's automatically generated by adding /feed/ in the search query URL. But I am not sure how to alert users when new posts appear under the RSS feed, and when to fire the alert (i.e. once a week), and how to display all new posts in an email template.

Any help is greatly appreciated!