How can I create a more transparent, thinner bottom area of smoke, then rising higher to be much thicker and opaque? It will not need to be animated.
Use a Gradient texture to control the mix between the smoke density and a transparent shader.
You can use generated coordinates and a mapping node to control the placement. In this case note the rotation on the $Y$ axis.
After the Gradient texture you can use color ramp to control the placement for the transparency.