A swirl is a very fine scratch cut into the surface of your clearcoat layer. The reason why you see swirls to begin with is because of light refraction, like through a prism. When you have something that breaks the continuity of a smooth surface, light refraction is altered and it's easier to see the defect. When you have a dark basecoat paint, such as (sigh) black, the contrast is greater and the swirls stand out more.
There are two ways to handle swirls.
Apply a glaze, such as Meguiars 07. A glaze will dry clear and fill in the swirls on the clearcoat, thus restoring the continuity of your smooth clearcoat layer and correcting light refraction. You are not actually removing the swirls, you're hiding them. This is what you typically pay for when you have your car detailed at the car wash and it costs about $60 and takes only takes about 2 hours. Glaze will wear off over a few washings and the swirls will re-appear.
Correct the paint. In this space, it would take too much time to explain how to properly correct the paint on your car, but this is the method that involves abrading the clearcoat paint layer to level the defects. Paint correction can be done by hand, but best results are achieved with a dual action machine polisher. While this is a permanent swirl removal process in that it removes the existing swirls, it won't prevent new swirls from forming unless you take precautions on how you wash your car, dry your car, and maintain the paint following a full paint correction.
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