Thanks for being helpful, but this isn't quite what I meant. I guess I need to be more specific.

I am making these textures for use in a program called Universe Sandbox, which takes mercator projections and maps them onto spheres. As a consequence, to render my polar caps I need white areas at the top and bottom of the image with a fractal-looking border, which gradually have less white the farther they get from the top and bottom edges, but do not lose opacity, only size and frequency of white areas.