This thesis addresses the planetary-scale mixing of tracers along
isentropic surfaces in the extratropical winter stratosphere.
The primary goal is a more fully quantitative
understanding of the mixing than is available at present.

The general problem of representing eddy mixing in a one-dimensional
mean representation of a two-dimensional flow is discussed.  The 
limitations of the eddy diffusion model are reviewed, and alternatives
explored.  

The stratosphere may, for some purposes,
be viewed as consisting of relatively well-mixed
regions separated by moving, internal transport barriers. 
Methods for diagnosing transport across moving surfaces,
such as tracer isosurfaces, 
from given flow and tracer fields are reviewed.

The central results of the thesis involve diagnostic studies of
output from a shallow water model of the stratosphere.  It is
first proved that in an inviscid shallow water atmosphere
subject to mass sources and sinks, if the mass enclosed by a 
potential vorticity (PV) contour is steady in time, then the 
integral of the mass source over the area enclosed by the contour
must be zero.  Next, two different approaches are
used to diagnose the time-averaged transport across PV contours in the model
simulations.  The first is the modified Lagrangian mean (MLM)
approach, which relates the transport across PV contours to
PV sources and sinks.  The second is called ``local gradient reversal'' 
(LGR), and is similar to
contour advection with surgery.  

The model includes a sixth-order hyperdiffusion
on the vorticity field.  Except 
in a thin outer ``entrainment zone'', the hyperdiffusion term has only a
very weak effect on the MLM mass budget of the polar vortex edge.  
In the entrainment zone, the 
hyperdiffusion term has a significant effect.  The LGR results 
capture this behavior, providing good quantitative estimates
of the hyperdiffusion term, which is equivalent to the degree of
radiative disequilibrium at a PV contour.  This agreement
shows that the main role of the hyperdiffusion is to 
remove filaments.  It is
argued that these results do not depend on 
the details of the small-scale dissipation.

\begin{sloppypar}
 Using a more direct type of trajectory-based calculation, the
``transilient matrix'' for the shallow water model flow is 
constructed.  The matrix is used as the basis for a one-dimensional
chemical transport model of
the two-dimensional shallow water flow.  A highly idealized
representation of (true) latitude-dependent chemistry is included.
The one-dimensional model represents the two-dimensional
model reasonably well, but is surprisingly
insensitive to some details of the transilient matrix. 
The transilient matrix calculations also show, as expected, 
that the model polar vortex is extremely isolated from its exterior.  
\end{sloppypar}

The various different diagnostics, taken together, allow a comprehensive
description of the Lagrangian circulation in the model's winter extratropics
to be composed, including the relationships between parcel trajectories
and PV contours in different flow regions.