Mathematical Diversion - 4 1.
Notation here is conventional.
Terms 1-3 are pretty obvious. Term 5 results from cross-differentiating the vertical advection terms of the momentum equations, and represents the tilting of horizontal vorticity into the vertical. If the flow starts out without any vorticity in three dimensions, then this equation shows that the only way to create vertical vorticity is through the solenoid and friction terms. Hence, they are the only true source terms. There are two more such equations for the other two components of the vorticity vector, of course.