where ω is the growth rate, k is the wave number, a is the radius of the bubble cylinder, T is the surface tension, Γ is the circulation, and K0 and K1 are Bessel functions. When ω is positive the bubble is stable due to circulation and when ω is negative, surface tension effects destabilize it and break it up.[8] Circulation also has an effect on the velocity and radial expansion of the bubble. Circulation increases the velocity while reducing the rate of radial expansion.