In particular, what happens if the Sawyer–Eliassen equation cannot be solved? I know that a condition for the Sawyer–Eliassen equation to have a solution is $F^2 N^2 - S^4 >0$, which means that the potential vorticity is positive.
I was wondering what is the physical implication of this condition on frontogenesis? Are there situations where fronts cannot be created, when potential vorticity is negative? Does that occur somewhere?
A secondary question would be the link with symmetric instability, which requires if I am not mistaken that the absolute vorticity (which is more or less potential vorticity) must be negative (in the northern hemisphere). I do not understand, then, how symmetric instability can arise when the Sawyer–Eliassen equation can be solved. Instability should occur in fronts since it is the driving phenomenon for the creation of precipitation bands!