These boundaries offer a closure in the budgets of vorticity and potential vorticity since the advective fluxes are exactly zero across material curves and all advective transport can be viewed through the exchange and topological connections of distinguished material regions. Non-advective fluxes then act across the Lagrangian boundaries. We see that there are multiple configurations in which there is limited transport across the Eulerian boundaries but the Lagrangian boundaries show that the entrainment does not reach the core. Since the Lagrangian boundaries can be located in flows with arbitrary time-dependence, they offer a generalization of the marsupial paradigm for tropical cyclogenesis.
The entrainment of these manifolds terminates in a limit cycle at the periphery of the core. The boundaries are further enhanced by the residual three-dimensional structures that are left behind from rotating convection. The accumulation of these structures forms an additional boundary closer to the cyclone core, a shear sheath that bounds the region of highest rotation. This boundary is observed as the maximum radial gradient of the Okubo-Weiss parameter integrated along particle trajectories (Lagrangian OW). This boundary, once formed, is a complete barrier around the inner core, and prevents any further transport from occurring, including dry air that would disrupt the cyclone, or mergers which would increase the circulation. Combining these ideas of Lagrangian boundaries for the outer flow and inner core, the question of development versus non-development becomes (i) whether the lateral transport of dry air is sufficiently limited so that a shear sheath can be established and (ii) whether convection is sufficient to stretch the small-scale vortices vertically, increasing their mean vorticity.
The existence and impermeability of Lagrangian boundaries is important for distinguishing developing versus non-developing disturbances. Since these boundaries are frame-independent, they can be found in an automated manner, and are thus useful for the evaluation of global weather models. Examples of Lagrangian boundaries are shown for recent developing and non-developing cases.