Lagrangian Trajectories At The Outflow Of Tropical Cyclones

Recent observations from the upper levels of tropical cyclones show the development of a Low engulfed by a larger High. The impact of such a geopotential map on the dynamics of the outflow is studied by analysing the Lagrangian trajectories of air parcels. The geopotential is modelled by the hydrostatic sum of two, oppositely signed, Gaussians: a warm core and a near-surface low that together reproduce the relevant features of the observed tropical cyclones. Assuming a time-independent and axisymmetric geopotential map, the dynamics is described by an integrable, two degrees-of-freedom, angular momentum conserving, Hamiltonian system with a single non-dimensional parameter. The steady states of this system bifurcate when the geopotential amplitude increases above a threshold where the single elliptic fixed-point becomes hyperbolic surrounded by a pair of elliptic fixed-points. A gradient balance prevails near the elliptic points while an inertial balance occurs at the radius of maximum geopotential, where the hyperbolic (unstable) point is located. The divergence of trajectories around this hyperbolic point implies the existence of strong horizontal wind divergence. A gradient non-balance region emerges in the steady state (radius, geopotential amplitude) map and this region expands with the increase of geopotential amplitude which implies that higher angular momentum values are required for trajectories to transition from high to low. A transformation of the (radius, radial velocity) variables to action-angle variables yields an approximate expression for the tangential drift (long-time average of the tangential speed) that estimates the tangential winds. The analysis is extended numerically to the case of warm core offset, where the two Gaussians have different centres.