No. The only way that makes sense is that to apply, they have to personally mail you or another admin a handwritten postcard and pray that you like the contents of the photo enough to let them join.
Joking aside, I was invited to a private group through referral. I didn't join, because it wasn't my scene, but I know they exist. I get the suspicion that what we currently have going on, where someone can't post links until they've filled a circle or have been promoted, is part of the reason why the site is so slow in traffic these days. I try to counter act that by promoting users I come to discover clearly aren't bots, but some people just don't stick around that long.
While referral makes some sense, I'm afraid it would create an echo chamber effect here, because that's how referral groups tend to operate. Application also makes sense, but I'm afraid people would get the impression that this place is elitist and/or unwelcoming. Either way, I'm worry that maybe both would tamper down on site traffic even further.
Those are just my two cents. Ultimately this is your site and I respect your decisions in how to run it. To be fair, I've signed up for this place four times already, so obviously you're something right. :)