GIMP :: Curve A White To Black Gradient

What I'm trying to do is curve a white to black gradient along a windscreen wiper smear for a rail engine cab window I need to texture. I'm building the cab for an engine in RailWorks and to get the wiper to clear raindrops from the screen in time with the wiper motion the texture needs a gradient applied to it from the wiper rest position to full up position.   

In the picture the actual wiper smear is all black. What I need to do is have it all white from the vertical part of the smear or the wiper rest position, to all black at the maximum part of its stroke. This means the gradient will be a lot shallower on the outer edge of the arc and a lot steeper at the inner edge of the arc.