Stirring and Mixing in the Stratosphere

1. Introduction
Transport and mixing play key roles in determining the distribution of ozone, and other important trace gases, within the stratosphere. In particular, understanding/quantifying the mixing of different air masses is important for understanding the observed ozone depletion. In this study the mixing processes in the lower stratosphere are examined using a combination of high-resolution trajectory calculations, in-situ trace gas observations, and simple mathematical models.

2. Fine-scale tracer transport
The wintertime stratospheric flow is characterized by a strong, cyclonic polar vortex surrounded by a quasi-two-dimensional turbulent region (known as the ``surf zone'') which extends to the sub-tropics. During large amplitude wave events (so called ``Rossby wave breaking'' events) filaments of vortex (and tropical) air are entrained into the surf zone, producing a sharp-edge polar vortex surrounded by a ``sea'' of filamentary structures. This is illustrated in the figure below.
[Click on highlighted images for larger view.]

Figure 1: High-resolution depiction of the tracer distribution in the northern hemisphere lower stratosphere on 28 January 1992. The image was generated using a high-resolution contour-following technique (``contour advection'' (CA); Waugh and Plumb 1994 , Norton (1994)) together with observed wind fields. The contours were initialized 12 days earlier as potential vorticity contours from low resolution meteorological analyses.


3. Trace gas observations
How believable are the small-scale features in trajectory calculations, such as in Fig. 1, which use low resolution (in time and space) wind fields? In recent years high-resolution measurements of trace gases have been made during NASA aircraft campaigns. These measurements enable the reality of the fine-scale features to be assessed. Comparison of the simulations with these measurements show remarkable agreement. Three examples are shown below.

Figure 2: Comparison of simulation of the tracer distribution at 0 UTC on 24 January 1992 with aerosol measurements. (a) tracer distribution from a 8 day CA calculation using observed winds (Fig. 1 shows distribution after 12 days; using different color scheme). (b) vertical profile of aerosols measured aboard DC-8 aircraft (flight path is solid line in (a)); the 450K line corresponds to the horizontal cross-section shown in (a). Extra-vortex (high aerosol) air was observed inside the vortex (time = 23 and 25 hr) and a narrow region of vortex (low aerosol) air was observed outside the vortex (between time = 26 and 27 hrs), in good agreement with the simulation in (a). (See Plumb et al., 1994 for details.)



Figure 3: Comparison of simulated tracer distribution and tracer measurements for 6 January 1992. (a) tracer distribution from 11 day CA calculation using observed winds. (b) in-situ measurements of N2O and CH4 along the northbound leg of the ER-2 flight path (solid line in (a)). Consistent with CA calculations low values of tracers representative of vortex air are sampled at northern end of flight leg. (See Waugh et al., 1994 for details.)



Figure 4: CA calculation showing the evolution of vortex air following the break up of the Arctic vortex in April 1993. The circles on plots for 30 April, 1, 6, and 7 May are the location where tracer values representative of vortex air where measured aboard the ER-2 aircraft. There is good agreement between the location of the filaments of ex-vortex air in the simulation and the locations where vortex air was observed (See Waugh et al., 1997 for details.)

The above (and other) comparisons indicate that the fine-scale features generated in high-resolution trajectory calculations are realistic, and these calculations can be used to examine the stirring and mixing occurring within the stratosphere.

3.2 Tracer-tracer relationships
As well as verifying the structure shown in numerical simulations, the aircraft measurements can be used to provide insight into the small-scale mixing processes. Points on scatter-plots of two long-lived tracers (trace gases whose chemical lifetime are longer than transport time scales) form smoothly varying curves. Mixing of two air parcels with distinct tracer values can produce an ``anomalous'' straight line (so called ``mixing line'') on the scatter plot, as shown in Figure 5(a). Therefore, tracer-tracer plots can be used to identify measurements in partially-mixed air masses. Anomalous mixing lines are observed in scatter-plots of tracer measurements from several different ER-2 flights. An example for 7 May 1993 flight is shown in Figure 5(b). This data is used in the analysis below.


Figure 5: (a) Schematic diagram of the effect of mixing on tracer-tracer scatter plots. The solid curve represents the ``standard'' correlation curve of two long-lived tracers. The triangle represents the result of total mixing between the discrete airmasses labeled A and B (mass of B is larger than A). Partial mixing between the discrete airmasses produces an anomalous mixing line (dashed line). (b) Scatter-plot of CFC11 versus N2O for data from 7 May 1993 flight from the SPADE campaign. The curve corresponds to a fit from previous ER-2 campaigns.}

4. Quantification

4.1 Stretching Rates
The stretching and folding occurring within the surf zone leads to an exponential decrease with time in the horizontal scale of tracer features. The characteristic time scale, or equivalently horizontal strain rate S, can be estimated from the exponential lengthening of material contours. Calculations for lower stratosphere during winter/spring yield S approximately 0.2 day-1.

4.2 Vertical cascade
Although the large-scale flow is quasi-horizontal, the vertical scale of tracers cannot be neglected, and in general there is also a cascade in the vertical scale. In the case of a steady strain flow, the combined effect of horizontal strain and vertical shear leads to exponential decrease in vertical scale at the same rate as that of the horizontal scale (so that the aspect ratio of horizontal to vertical scales remains constant with time). ...

This relationship also holds in time-dependent "turbulent" flows if the horizontal strain and vertical shear both vary on a time scale at least as long as the inverse strain rate (Haynes & Anglade 1997). Furthermore, calculations for wintertime stratosphere flow show that the aspect ratio is around 250:1. In other words, the large-scale stratospheric flow tends to produce sloping sheets of tracers that are close to horizontal.

4. 3 Diffusivities
The cascade in scales due to the large-scale flow will be halted by diffusive (or other small-scale mixing) processes. As the characteristic aspect ratio of scales in the stratosphere is large the mixing will occur first in the vertical. However, as vertical and horizontal scales tend to stay in same aspect ratio the vertical diffusion will dissipate horizontal scales as well as small vertical scales, and a vertical diffusion D causes an effective horizontal diffusion Dh = \alpha2 D .

The key components of stratospheric mixing can be captured in a simple one-dimensional advection-diffusion model with two parameters: the horizontal strain rate S and effective horizontal diffusion Dh. The appropriate value of S can be estimated using trajectory calculations, but what is an appropriate value of Dh for the lower stratosphere? Molecular diffusion sets a lower limit on D (and hence Dh as horizontal and vertical scales are coupled), but other processes, such as patches of three-dimensional turbulence, are thought induce mixing before the large-scale flow reduces features to molecular scales.

An estimate of Dh can be obtained using high-resolution aircraft observations of trace gases together with trajectory calculations and the one-dimensional advection-diffusion model. From an analysis of May 1993 data we estimate that Dh = 103 m2/s, see Waugh et al., 1997 for details. Assuming an aspect ratio of 250 , Dh = 103 m2/s corresponds to a vertical diffusion D = 0.015 m2/s (compared to D =10-4 m2/s for molecular diffusion).

Using the above estimates the 1D advection-diffusion model indicates that a filament (with initial width 500 km) does not change for the first 7-8 days, changes rapidly in the next 15 days as the filament is stretched out and mixes with the background air, and is completely mixed with the background air within a month (solid curve in Fig. 6).


Figure 6: Temporal evolution of maximum \sigma for various values of S and D. Initially \sigma is 1 inside a 500 km wide filament and 0 outside.}


5. Seasonal Variations

Although, as shown above, there is rapid stirring and mixing during winter/spring there are indication that these processes have a large seasonal variations. In particular, during summer months there is reduced wave activity (reduced vertical propagation from troposphere) and weaker stirring (S = 0.1 day-1). Also, in-situ tracer measurements indicate that structures can remain unmixed for at least 2 months during the summer. Even allowing for the reduced S this implies smaller Dh in advection-diffusion model (dashed and dotted curves in Fig. 6).

6. Summary

High-resolution numerical simulations indicate that the winter stratosphere comprises of a sharp-edged polar vortex surrounded by a quasi-two dimensional turbulent region. In-situ measurements of trace gases show small-scale features consistent with the filamentary structure in these simulations, indicating that these simulations can be used to examine the mixing within the stratosphere.

Analysis of trajectory calculations together with in-situ tracer data indicates that during winter/spring that the stretching rates are around 0.2 day-1, the aspect ratio of horizontal to vertical scales is around 250, and that the vertical diffusivity is around 0.015 m2/s. This suggests that the time scale for horizontal scales of 1000 km to be reduced to mixing scales (50 m in the vertical, 10 km in the horizontal) is around 10 to 15 days, and that complete mixing occurs within a month.


Darryn Waugh