Effective diffusivity calculations

Atmospheric changes, such as ozone depletion and global warming, are of ever-increasing concern. International agreements to limit the anthropogenic emissions of gases that cause such changes are politically viable only with hard scientific evidence and predictive capability to support their objectives. It is therefore imperative that scientists continue to seek a better understanding of the processes acting in the atmosphere. The chemical processes are clearly important, but also crucial are the processes that transport chemical species from one location to another. My research focuses on quantifying the transport processes. In particular, I have examined the semi-permeable barriers to transport, i.e. regions of weak mixing that separate well-mixed regions of distinct chemical characteristics.

The atmosphere is, on the large scale, stably stratified and consequently `vertical' transport occurs on timescales of weeks--months, whereas `horizontal' transport occurs on timescales of days. More precisely this fast `horizontal' transport is along surfaces of constant entropy or isentropic surfaces (see figure 1). It is useful, therefore, to consider only the two-dimensional flow along isentropic surfaces. Furthermore, the flow along such surfaces is approximately incompressible.

Figure 1: The structure of the lower atmosphere. The purple lines represent isentropic surfaces. The arrows denote mixing. The red lines represent barriers to transport along isentropic surfaces. The approximate altitude at which passenger jets fly is shown . The green region (the troposphere) typically has low ozone and high water vapour mixing ratios. The blue region (the stratosphere) typically has high ozone and low water vapour mixing ratios.The blue shaded area represents the region in which ozone loss occurs in spring.

The transport of chemical species along isentropic surfaces is not spatially homogeneous. The observed spatial distributions of chemical species show apparently well-mixed regions between which there are rather sharp transitions in chemical concentrations. This suggests that transport between different well-mixed regions is limited by partial barriers (figure 1). Such barriers are not geographically fixed, but are continually moving and may change position substantially over a season. The permeability of the barriers also changes with time. To understand atmospheric transport for the calculation of long-term chemical change, it is clearly essential to identify these barriers and quantify their permeability.

Quantifying the permeability of a barrier to transport in atmospheric flows, or even in simple chaotic advection flows, is not a straightforward exercise. Since the barriers are not geographically fixed, the difficulties of quantifying and identifying them are inextricably connected.

The traditional approach taken to investigate transport in the atmosphere is to use observed winds to advect particles in numerical simulations, and then to calculate stretching rates measured by particle separation (Norton, 1994; Waugh et al, 1994; Chen, 1994). Regions where the stretching rate is comparatively small are associated with barriers. Diagnostics based on this approach have given some insight into the transport and mixing structure of the lower atmosphere. However, they have failed to identify some of the barriers whose existence is inferred from chemical distributions (Bithell and Gray, 1997). In particular, the traditional approach appears to fail when the timescale for a particle to cross a permeable barrier is less than the timescale necessary to define a stretching rate. The limitations of this approach are discussed further below.

I have used an approach to diagnosing the transport structure of a flow developed by Nakamura (Nakamura, 1996).

Barriers are generally associated with regions of weak mixing, separating regions of strong mixing. In a region of strong mixing a tracer will develop a complex geometric structure. The approach is based on using the effective diffusivity, a quantity that characterises the geometric structure of a tracer (Nakamura, 1996), to deduce the mixing ability of a two-dimensional, incompressible flow with velocity field , u(x,y,t).

Consider a tracer whose amount per unit mass of air is c(x,y,t), that evolves according to an advection-diffusion equation with constant diffusivity, , i.e..
If the area, A(C,t) within a tracer contour, c(x,y,t)=C is used as new co-ordinate, the advection-diffusion equation reduces to the form,

where , the effective diffusivity, is defined by

here is the average over the area between adjacent tracer contours. The quantity includes the effects both of stirring (through the term) and homogenisation (through the term). Furthermore it can be related to the actual contour length, C, since,

here is the line average around a tracer contour. The effective diffusivity is therefore largest where the tracer contours are longest, i.e. where their geometric structure is most complex.

My approach has been to quantify the transport and mixing structure of a flow, by calculating associated with a test tracer field. A computer simulation is used to follow the evolution of the tracer. For atmospheric flows there is evidence that the large scale flow gives the dominant contribution to advection.  Therefore, comparatively low resolution wind data can be used to predict high resolution tracer fields. The integral expressions above are then evaluated from the tracer field. Large values of  are associated with mixing regions, small values with barriers.

By calculating from tracer fields generated by chaotic advection flows, I have demonstrated that the  diagnoses the mixing ability of the flow itself, being essentially independent of the particular test tracer used in the calculation. I have used chaotic advection flows as the basis for the comparison of effective diffusivity with other measures of transport and mixing.

I have calculated the effective diffusivity for atmospheric flows, defined by wind data from weather forecasting centres. This has revealed a global picture of the transport and mixing structure of the lower atmosphere, unifying and clarifying results of previous particle-based studies.

Figure 2: Effective diffusivity for one isentropic surface in the stratosphere showing the Arctic and Antarctic polar vortices. Weak mixing implies the presence of a barrier. The pictures are a) Antarctic, 31st August 1997, b) Arctic, 28th February 1998, times when ozone depletion is commencing and the ozone hole is beginning to form. Red indicates strong mixing, blue weak mixing.

The so-called `ozone holes' are associated with strong vortices that exist in winter and spring in the stratosphere over the poles. A leaky barrier at the edge of the vortices partially isolates the polar air, providing a unique environment in which chemicals such as CFCs can be converted to a form which, in the presence of sunlight (which returns to polar regions in spring), destroys ozone. The effective diffusivity show the complete vertical structure and seasonal evolution of the transport and mixing associated with both the Arctic and Antarctic vortices. In particular, the diagnostic provides a quantification of the transport across the leaky barriers at their edges, something which is critical for predicting ozone loss.  The results clearly show that the barrier at the edge of the Arctic vortex is more distorted and weaker than the Antarctic vortex (the 450K surface is shown in figure 2), which is consistent with less ozone loss.
 

The effective diffusivity shows that in the Antarctic the barrier at the edge of the vortex extends down into the troposphere, preventing activated CFCs from leaving the polar regions. Conversely, in the Arctic the barrier does not extend so far down and hence activated CFCs can leave the sub-vortex region and, in the presence of sunlight in mid-latitudes, cause widespread ozone destruction over populated regions.

Transport across the notional barrier between the lowermost stratosphere and the troposphere - the tropopause - constitutes the main mechanism for the removal from the stratosphere of chemicals involved in ozone destruction, such as CFCs and those emitted from aircraft. The effective diffusivity presents a complete characterisation of the structure of the barrier associated with the tropopause, where previous attempts, using particle-based methods, have failed to identify a barrier at all. Our results also show the seasonal variation in strength of this barrier, which is seen to be weakened by summer monsoon anticyclones. The most fundamental and useful definition of the tropopause is, arguably, this transport barrier. We have compared our results with the more traditional definition based on potential vorticity.

Our results have also shown the structure and seasonal evolution of transport into and out of the tropical stratosphere, thereby clarifying the picture that has been obtained from observations of chemical and volcanic debris distributions. Furthermore, we have shown that some mixing occurs in the lower stratosphere of the summer hemisphere, a region which has up to now been regarded as rather quiescent.

In summary, we have demonstrated that effective diffusivity is a powerful new diagnostic which should be added to the repertoire of techniques for analysing atmospheric flows, and by calculating effective diffusivity we have obtained  a number of new results which have important implications for predicting atmospheric chemical change.