I wouldn't go right to a "sit down and talk meeting." Start by seeing if some pointed sentences in the moment get the message across. Next time she does it, raise your eyebrows and say, "Okay, Julie. I know it's involuntary but there no reason to make a scene about it." Hopefully she will get your point without you having to have a come-to-deity about it.
actually, if she has enough warning that she can lift her left, she could probably control it long enough to leave the room.
But I like this approach, because there's no sense in making a big deal about whether she should fart around other people. Focus on the grossness of making a scene.
And having had to have convos like this now and then, I'll say this:
Make it short.
Make it direct.
Don't allow for discussion.
Make it an order.
Say, "I need you to change something. You lift your left and make a bit of a scene when you are passing gas. Don't do this anymore. The best thing to do is to leave the room. If you can't, and I recognize how disruptive it can be to have to leave the room, then don't make a scene about it."
If she tries to make it discussion, etc., say, "I don't want to have a discussion about this. This is an order, actually. I don't generally come along and be a bossy boss, but this time, it's an order."
Yes, traditional etiquette says you never, ever say "excuse me" for farts. You do for burps, but farts are considered -so- disgusting that one never, ever mentions them.
If I felt I wanted to give a reason for my "direct order," that's what I'd say. And so if you're not, in polite society, even supposed to mention them in apology, it is certainly not polite to make them a joke and call everyone's attention to them.
Be short. Be direct. Be firm. Don't allow discussion.
You don't need her to agree with you, or to persuade her. Oh, sure, that's a useful approach sometimes. But....given her personality, and the type of joke she is making, I don't think it will be effective so don't even bother.
Think about it this way: She -knows- this is impolite. So channel your Inner Day-Care Worker (to get the calm authority you need), and simply tell her, "this is not the way I want things to be in this office. Please keep your farts to yourself, as best you can."
I like Goog's script as well.