Transition to turbulence in an elliptic vortex

We study the three dimensional instability and nonlinear growth of the two dimensional flow described by the stream function Psi = (A sin b1 x sin b2 y)/(b1(exp 2) + b2(exp 2)) where b1 = pi/L1, b2 = pi/L2. This is a swirling flow in a box which is bounded by 0 less than x less than L1, 0 less than y less than L2 and is infinite in the z direction. This flow is a solution of the Navier-Stokes equation with A = exp(-v(b1(exp 2) + b2(exp 2))t) which slowly decays. We seek a viscous solution which starts near this one and slips along but does not penetrate the bounding walls. The vorticity of the basic flow is w(sub z) = A sin b1 x sin b2 y which has maximum value A at the center of the box and drops to zero at the boundaries. We can think of the resulting flow as that of a captive vortex.