Hello

How can I add noise only on the black area.

Thanks first.

Your file is kind of a mess. bgeo's necessary to get the scene setup aren't included in the zip. If I were you, when posting something like this, I'd simplify as much as possible, or remove any extraneous stuff, so we don't need to search around and try and find where you are having the problem, because most people won't do that.

What I would have done in your situation is try and create a ramp along the edges of your black area, so that there is a falloff from white to black. Invert the colors so the black area becomes white and the white area becomes black. When you have white in your center area, you can add turbulent noise to the position multiplied by the point color, and as long as you have a gradient of fall off between the two colors, it should look smooth.