Local sound-speed estimate for ring-flow stability ($n^2<\frac{m{R}^2{\mu }_0}{{\hslash }^2}\frac{1-a}{(1+a{)}^2}$)

Combining zero-current bulk densities with quantized circulation gives an approximate winding bound. If equal halves have densities ${\rho }_0(1-a)$ and ${\rho }_0(1+a)$, $0<a<1$, the more restrictive local condition is $n^2<(m{R}^2{\mu }_0/{\hslash }^2)(1-a)/(1+a{)}^2$. A self-consistent finite-current calculation must also allow kinetic pressure to change those densities.