the above equations, $r_i$, $r_o$ are the radii of the inner and outer cylinders, respectively. In what follows, we assume that the inner cylinder rotates with a constant angular speed $\om_i$, while the angular speed of the outer cylinder changes periodically with $z$ around the average value of $-\om_i$: $\om_o=\om_i(-1+\dt \sin(\la z))$, where $\la$ and $\dt$ are the wavenumber and amplitude of oscillations, respectively. Qualitatively same results (differing in the exact expression for the adiabatic invariant and, consequently, in specifics of the resonance equations) can be obtained if, instead of the varying $\om_o$, the outer cylinder has a varying radius. Similarly, we could consider a flow inside a torus-shaped container instead of the annulus.

To introduce dimensionless variables, we rescale the time by $\om_i$ and the distances by $r_i$: $$\ovl{t} = t \om_i, \quad \rho=\frac{r}{r_i}.$$ From now on we assume that all the variables are in the dimensionless form and the dot denotes the derivative with respect to $\ovl{t}$. For the numerical simulations presented below the following set of parameters was used: $$r_i = 0.2, \; r_o = 1.0, \; \eta = 0.2, \; \dt = 0.4, \; \la = 2\pi.$$ In what follows, we assume periodic boundary conditions in 