What you refer to is circular color banding. It's usually added by the camera to hide sudden transitions.
It also happens in Photoshop-Gradients.
If what you posted above is your original shot, then let me guess: Your shot is a jpg. In your case shooting RAW to prevent that the camera manipulates the image would have been the way to go.
There's not much if anything you can do to "rescue" this shot if this is the original jpg.
rings have to do with how your saving it. it's like it was saved for web and it was simplied in the sky. usually you'll see banding on a cheap lcd screen. but they've come a long way since then. so i'm sticking with saving and compressing issues. i'm seeing a bit of colored noise in there too.
The gradient(s) are often referred to as "moire", and I recall a You Tube video on how to fix things. I do not remember if the video was specifically addressing Photoshop gradients, or those in pictures.
Using then RAW file really helped the image. However even then when I went to adjust the image in the brightness settings I ran into the "moire" or banding, however much less noticeable. Maybe the fog had something to do with it, as it was quite heavy this morning. I added a texture overlay to try to help cover the very, very slight banding that occurred when I decided to go ahead and adjust the brightness anyway... and I kind of like it now actually!
This little exercise you just went through really demonstrates the power of the RAW file and why it's the only way to shoot. I hope more people see this and realize that this is the way to capture images.
What was your ISO and what was your camera/lens used?
An image like that has a very narrow exposure range, just a couple of sharp peaks on a histogram, I would think. If you have tried to spread out the exposure to fill the histogram (i.e moving in the left and right sliders) then you would be stretching the tones out, which could do nasty things to them. It does appear under-exposed, which suggests that at the very least the exposure was pushed up a stop or so in processing the RAW version, which would also tend to create problems.