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5. Effective diffusivity

An alternative diagnostic for measuring the mixing properties of a flow is the effective diffusivity, as described in Winters & D'Asaro (1996) and Nakamura (1996), and applied to atmospheric problems in Nakamura & Ma (1997), Haynes & Shuckberg (2000a,b). The effective diffusivity has the advantage over the contour stretching rates described above of being a hybrid Eulerian-Lagrangian quantity that therefore avoids the problems encountered by purely Lagrangian techniques near relatively weak barrier regions. The effective diffusivity of a scalar field is proportional to the square of the equivalent length, L_e, of a contour enclosing the region where the scalar field is greater than a given value [see eg. Nakamura (1996) for details and definitions]. The equivalent length is in turn related to the actual material length, L, of the contour by the inequality L_e £,L with equality when the gradient of the tracer field is uniform (in magnitude) around the contour.

To compute L we use again the contour advection model, but this time include the surgery algorithm. This removes those scales below a certain threshold (in this case 20km) with the result that the contour lengths equilibriate after a few days, the generation of new fine-scale structures balancing the removal of others by the surgery. We then use the resulting contour length, normalized by the length of the equivalent latitude contour, as an approximation for the equivalent length L_e.

The results of five years of simulations, one six day simulation per month on the 350K isentropic surface, with twenty contour intervals per hemisphere spanning the PV values from (±) 0.4 to 8.4 pvu, are shown in Figure 3. Here, L_e is plotted as a function of equivalent latitude and time. The darker shades represent low values of L_e, and hence with areas of low effective diffusivity and low mixing rates. Both the seasonal variability and the interhemispheric differences are clearly visible. The former includes both the intensification of the tropopause as a mixing barrier in winter (cf Figure 2, identified by the darker shades, as well as the poleward migration in the late summer (cf Haynes & Shuckburgh, 2000, who found a similar pattern in the effective diffusivity calculated from particle integrations). The interhemispheric differences are characterized by weaker overall mixing rates in the SH (darker shades, corresponding to lower L_e) as well as a less well defined latitudinal migration of the tropopause with the annual cycle. This latter feature is consistent with the different characteristics of baroclinic activity between the two hemispheres. Finally, we note that similar interannual variability is seen in L_e as was seen in the stretching rates above, most notably lower values of L_e in the 1998 NH winter and higher values of L_e in the 1995 NH and SH winters.

Figure 3: Equivalent length of isocontours of PV, plotted as a function of equivalent latitude and time in the NH (top) and SH (bottom).

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