Making pseudo-polar plot in matplotlib

Question

I have been trying to make grids in the following shape.

For the grayed ⊃ shaped area (which is called magnetosheath in space physics), I divided it into several pieces, and naturally angular and radial direction would fit.

For example, in angular direction, I divided 0-10 degree, 10-20... 80 - 90 degree (and also to the opposite direction). And in radial direction, if I know the inner boundary r1(function of theta) and outer boundary r2(theta), I can divide it into 10 steps, like [r1,r1+(r2-r1)/10], and so on.

I modeled a parameter (plasma density) and I have a parameter value for each cell. Now I want to plot the parameters using matplotlib, something like following but under real shape of above figure.

However, I cannot use the normal matplotlib polar plot as this area doesn't have circular shape.

I think some kind of transformation from the polar coordinate system to this coordinate system is needed, but I couldn't find it. I mean, I can transform lattice points of this coordinates to cartesian one, but filling the grid in the cartesian coodinates would be an entirely different thing.