Unmixed islands in quasi-periodically-driven flows

Nested invariant 3-tori surrounding a torus braid of elliptic type are found to exist in a quasi-periodically forced fluid flow. The Hamiltonian describing this system is given by the superposition of two steady stream functions, one with an elliptic fixed point and the other with a coincident hyperbolic fixed point. The superposition, modulated by two incommensurate frequencies, yields an elliptic torus braid at the location of the fixed point. The system is suspended in a four-dimensional phase space (two space and two phase directions). To analyze this system we define two three-dimensional, global, Poincar\'{e} sections of the flow. The coherent structures (cross-sections of nested 2-tori) are found to each have a fractal dimension of two, in each Poincar\'{e} cross-section. This framework has applications to tidal and other mixing problems of geophysical interest.