Your original kitty is maintaining his/her place in your house as the dominant kitty. She’s basically slapping the other one down, saying This is house and my pet human, so stay in your place, I’m the boss. Your new kitty will be poking and prodding your original kitty to test the boundaries.
They are going to have scraps, even when they get used to each other they’re most likely to have scrag fights on a daily basis. We have three kitties, two from the same litter, and the other is their nephew. They fight and carry on often, pick fights etc, and like augustlan’s kitties, they’ll end up grooming one another and then fall asleep in a big kitty heap.
There are times when they are genuinely angry with one another, and then the fight will be on. That is definitely the time to intervene. When the fur is flying, and they are screaming at one another, that is when you need to tell them that it’s not on. We’ve found that raising our voices, and loudly, sometimes to a shout, will cause them to retreat. If it doesn’t, then you’d need to separate them for a couple of minutes. That is all it should take. You’ll soon be able to tell by their tone of voice which kind of fight they’re having (if you don’t know already). If you don’t want to physically get in between them, or pick one of them up, squirting them with a water sprayer should quickly break them up.