Hyperbolic lines and the stratospheric polar vortex.

Abstract

The necessary and sufficient conditions for Lagrangian hyperbolicity recently derived in the

literature are reviewed in the light of older concepts of effective local rotation in strain coordinates.

In particular, we introduce the simple interpretation of the necessary condition as a constraint on the

local angular displacement in strain coordinates. These mathematically rigorous conditions are

applied to the winter stratospheric circulation of the southern hemisphere, using analyzed wind data

from the European Center for Medium-Range Weather Forecasts. Our results demonstrate that the

sufficient condition is too strong and the necessary condition is too weak, so that both conditions fail

to identify hyperbolic lines in the stratosphere. However a phenomenological, nonrigorous, criterion

based on the necessary condition reveals the hyperbolic structure of the flow. Another still

nonrigorous alternative is the finite-size Lyapunov exponent FSLE which is shown to produce

good candidates for hyperbolic lines. In addition, we also tested the sufficient condition for

Lagrangian ellipticity and found that it is too weak to detect elliptic coherent structures ECS in the

stratosphere, of which the polar vortex is an obvious candidate. Yet, the FSLE method reveals a

clear ECS-like barrier to mixing along the polar vortex edge. Further theoretical advancement is

needed to explain the apparent success of nonrigorous methods, such as the FSLE approach, so as

to achieve a sound kinematic understanding of chaotic mixing in the winter stratosphere and other

geophysical flows.