question about supercell generation in Holton's book

[slow_learner]

Hello everyone. I have recently started studying the book An Introduction to Dynamic Meteorology by James R. Holton to improve my understanding of supercell convective storms. 

However, there are some parts of it that quite confuse me and I was wondering if someone could help me to understand this.

In the page 300, Holton linealizes a flow consisting of a single convective updraft in a basic state westerly flow which depends on z alone by making the vorticity w= j*du/dz+w'(x,y,z,t) and the speed of wind U= i*u+U'(x,y,z,t), being u the mean westerly flow. My question is: what do the terms w' and U' represent? I think that they are a first order terms on a Taylor expansion, but then, Holton writes the curl U x w =i*w'du/dz +j*uç', being ç the vorticity in the z axis, this debunks what I have written, since it seems that the module of the derivate of the vorticity (w') and the derivate of the vorticity on the z direction (ç) coexist. 

So, can someone please tell me what these terms, w' and U' represent?

Also, on the resulting equation, dç'/dt = -u*dç'/dx + dw'/dy * Du/Dz, beinf d/di a partial derivative and D/Di a total one, I don't understand the difference between w' and ç'.

 

Thanks in advance for your answers.

Edited May 31, 2018 by slow_learner