Cross-jet transport of passive scalars in a kinematic model of the meandering laminar two-dimensional incompressible flow which is known to produce chaotic mixing is studied. We develop a method for detecting barriers to cross-jet transport in the phase space which is a physical space for our model. Using tools from the theory of nontwist maps, we construct a central invariant curve and compute its characteristics that may serve as good indicators of the existence of a central transport barrier, its strength, and topology. Computing fractal dimension, length, and winding number of that curve in the parameter space, we study in detail the change in its geometry and its destruction that is caused by local bifurcations and a global bifurcation known as reconnection of separatrices of resonances. Scenarios of reconnection are different for odd and even resonances. The central invariant curves with rational and irrational (noble) values of winding numbers are arranged into hierarchical series which are described in terms of continued fractions. Destruction of central transport barrier is illustrated for two ways in the parameter space: when moving along resonant bifurcation curves with rational values of the winding number and along curves with noble (irrational) values.