### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. The Method
- 3. The SLIMCAT Model
- 4. Meteorology of the Winter of 1999/2000
- 5. Results of Model Simulations
- 6. Application of Method to Measurements
- 7. Discussion
- 8. Summary
- Appendix A:: Details of the Inversion Procedure
- Appendix B:: Determination of Envelopes
- Appendix C:: Different Approximation to the Mixing Kernel
- Acknowledgments
- References

[1] A method is introduced for diagnosing mixing between the polar vortex and midlatitudes from tracer data. Tracers with different photochemical activities and lifetimes usually exhibit curved tracer-tracer correlation functions on an isentropic surface. The effect of mixing events is to populate the inner side of such a curve. Using simultaneous measurements of trace gases or model results, we exploit this process to calculate the distribution of recent origins in tracer space prior to such a mixing event. The method relies on both hemispheric and local data and is applicable to situations where mixing is nonlocal in tracer space. It is applied to measurements taken during the Stratospheric Aerosol and Gas Experiment (SAGE) III Ozone Loss and Validation Experiment/Third European Stratospheric Experiment on Ozone 2000 (SOLVE/THESEO 2000) winter campaign and to a chemical transport model simulation covering the same winter. In one of the cases studied, a vortex breakup and subsequent remerger of the vortex fragments in March 2000 results in significant diagnosed mixing. In a further example, an elongated filament shed off the polar vortex is characterized by anomalous composition. For the two high-altitude aircraft flights of the SOLVE campaign that probe the vortex boundary, a correspondence is found for mixing diagnosed in the measurements and in the model. Mixing timescales considered here are given by the life span of planetary waves, up to a few weeks.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. The Method
- 3. The SLIMCAT Model
- 4. Meteorology of the Winter of 1999/2000
- 5. Results of Model Simulations
- 6. Application of Method to Measurements
- 7. Discussion
- 8. Summary
- Appendix A:: Details of the Inversion Procedure
- Appendix B:: Determination of Envelopes
- Appendix C:: Different Approximation to the Mixing Kernel
- Acknowledgments
- References

[2] Tracer-tracer correlations are often invoked to assess transport and mixing in the stratosphere. Pairs of long-lived tracers generally form compact correlations (e.g., *Goldan et al.* [1980], as referenced by *Plumb* [1996], *Kelly et al.* [1989], and *Fahey et al.* [1990]). Different regions of the atmosphere are characterized by different such tracer relationships, e.g., the tropics and the midlatitiudes [*Volk et al.*, 1996]. A similar partitioning into different regimes emerges in an analysis of probability density functions of tracer mixing ratios [*Sparling*, 2000].

[3] An example of a tracer-tracer correlation plot is displayed in Figure 1. In this model result the parameter space displayed in Figure 1 spanned by the two tracers is populated to a very variable density. Like in the observations, polar-vortex, midlatitude, and tropical “canonical correlations” show up as ridges in the density plot, marked as “P”, “ML”, and “T”, respectively. *Plumb and Ko* [1992] discuss these correlations. They reflect the different compositions, ages, and photochemical histories of air parcels in those three regions of the atmosphere [*Schoeberl et al.*, 2000]. In the following, the (multidimensional) tracer parameter space is referred to as “tracer space”.

[4] The formation of different tracer-tracer correlations in different atmospheric reservoirs is a consequence of the presence of transport barriers [*Haynes and Shuckburgh*, 2000, Figure 1]. The robustness of these correlations has been used to quantify ozone depletion [e.g., *Proffitt et al.*, 1989], dehydration [*Kelly et al.*, 1989], and denitrification [*Fahey et al.*, 1990], provided however mixing processes can be discriminated against chemistry [*Rex et al.*, 1999; *Plumb et al.*, 2000; *Salawitch et al.*, 2002]. Explicitly calculating the local budgets of trace species, *Minschwaner et al.* [1996] and *Volk et al.* [1996] use the difference of tracer-tracer relationships between the tropics and the midlatitudes to give estimates of mixing timescales between the two reservoirs. An application of their method to the polar vortex would require a modification because of the greater importance of the seasonal cycle in high latitudes, compared to the tropics.

[5] Anomalous mixing has been observed in tracer data [e.g., *Waugh et al.*, 1997; *Michelsen et al.*, 1998]. In the wake of, for example, a major wave breaking event air parcels from different reservoirs are brought together and mixed, creating, by way of a linear superposition of their trace gas mixing ratios, a straight line across an otherwise thinly populated part of tracer space. However, such mixing lines are constrained by the approximately isentropic nature of transport and mixing, so mixing lines can only connect points on the canonical correlations that represent approximately the same isentropic level. Thus mixing is not only inhibited by the presence of transport barriers (such as the boundary of the polar vortex) but also by separations in potential temperature. Hence different “isentropic correlation functions” relating pairs of tracers may be expected to develop on different isentropic surfaces, reflecting the different average photochemical histories of air parcels on these surfaces [*Schoeberl et al.*, 2000, Plate 1]. An example is displayed in Figure 1, where in a part of tracer space a separation of the data points according to the isentropic levels of the underlying chemical transport model may be discerned (i.e., the “fingers” in the bottom right corner of Figure 1 corresponding to different isentropic model surfaces). Thus, alternatively to having tracer-tracer correlations depend on equivalent latitude (which effectively is the conventional approach), the set of correlative tracer data can be described equally by correlation functions parameterized with potential temperature. On the two hemispheres, tracers may develop different isentropic and canonical correlations, so here all further analysis is restricted to one hemisphere only.

[6] A contribution to mixing is brought about by nonuniform diabatic heating, leading to cross-isentropic dispersion of air parcels. *Sparling et al.* [1997] show that this dispersion is much smaller inside the polar vortex than in midlatitudes, suggesting that the polar vortex sees a relatively uniform subsidence with little interlevel mixing. Clearly, however, diabatic dispersion is a mechanism broadening correlative tracer sets, in competition with isentropic mixing. The topic is further discussed in section 7.

[7] If both chemistry and diabatic motion are slow compared to the adiabatic component of advection, then linear relationships ensue between pairs of such long-lived tracers [*Plumb and Ko*, 1992]. Timescales of mixing within atmospheric regions separated by transport barriers often satisfy this condition but timescales describing mixing between different regions may not. In such a situation one finds curved isentropic correlations (e.g., Figure 2). If in a thought experiment one takes two parcels from the same isentropic surface with mixing ratios of the two tracers matching the curved correlation and generates a linear superposition of their tracer mixing ratios (i.e., a mixture), the result is an air mass with anomalous tracer concentrations lying off the isentropic correlation curve, as in the studies cited above.

[8] The present paper will exploit this relationship between mixing and the distance of a tracer data pair from the corresponding “isentropic” correlation curve. It will introduce a methodology through which the process of the formation of mixing lines may be inverted to yield information about the distribution of recent positions of parcels in tracer space immediately prior to the mixing event that make up the observed or modeled mixed parcel. The timescales considered are hence those for isentropic stirring in the wake of planetary waves. The method depends on both local and hemispheric information, i.e., the local mixing ratios of a number of tracers at a point in time and space and their respective isentropic correlations.

[9] The method will be applied to the lower stratosphere during the 1999/2000 Arctic winter and spring. This period saw the comprehensive international measurement campaign Stratospheric Aerosol and Gas Experiment (SAGE) III Ozone Loss and Validation Experiment/Third European Stratospheric Experiment on Ozone 2000 (SOLVE/THESEO 2000), yielding a substantial amount of stratospheric tracer measurements. In the following, we will develop the methodology and show its application both to model results and to SOLVE measurements.

### 2. The Method

- Top of page
- Abstract
- 1. Introduction
- 2. The Method
- 3. The SLIMCAT Model
- 4. Meteorology of the Winter of 1999/2000
- 5. Results of Model Simulations
- 6. Application of Method to Measurements
- 7. Discussion
- 8. Summary
- Appendix A:: Details of the Inversion Procedure
- Appendix B:: Determination of Envelopes
- Appendix C:: Different Approximation to the Mixing Kernel
- Acknowledgments
- References

[10] Figure 2 depicts density plots of pairs of long-lived species taken from a model simulation, sampled on the 451 K isentropic model surface. The data form compact correlations with the most long-lived tracer computed by the model, N_{2}O. Two maxima of data density are evident in the Figure 2, defining the most typical mixing ratios in the polar vortex (about 115 ppbv of N_{2}O, labeled “P”) and in midlatitudes (about 250 ppbv of N_{2}O, labeled “ML”) at this isentropic level. For halon-1211 and CFC-11 the isentropic correlations with N_{2}O are positively curved. Here we adopt the viewpoint that data points on the inner sides of these curves result from mixing between high and middle latitudes; this will be formalized in the following. In the tropics (the top right corners of the plots) separate branches appear, constituting parts of the tropical canonical correlations. In computing the displayed isentropic correlation functions these branches are suppressed by excluding points south of 35° from the computation of the isentropic correlation envelopes (Figure 2). Accordingly, here the focus is on exchange between midlatitudes and the polar vortex only. The inner and outer sides of the isentropic correlation curves are marked “A” and “B”. In defining inner and outer sides the assumption is made that the sign of curvature does not change outside of the tropics; this is further elaborated in section 7.

[11] To formalize the above, let Ψ(ζ; Θ, *t*) = (Ψ_{1},…,Ψ_{p}) be the set of “correlation envelopes” bounding a set of correlative tracer data on the outer (B) side (see Appendix B for details). Here, ζ denotes the reference tracer N_{2}O, Θ is potential temperature, and the analysis time *t* accounts for a possible slow evolution with season. N_{2}O is chosen as the reference tracer because it has a good contrast across both the edge of the polar vortex and the subtropical barrier. There are *p* different tracers considered here. By definition the reference tracer satisfies Ψ_{1}(ζ) = ζ. Examples are displayed in Figures 2 and 3a. The Ψ_{i} describe an idealized, “unmixed” atmosphere where correlative data points of pairs of long-lived tracers form compact curves on an isentropic surface. The two lines depicted in Figure 3a illustrate two possibilities for isentropic mixing of points on a correlation envelope which can generate, by way of a weighted linear combination, any point on the mixing lines. For the measurement depicted, both lines are possible, as is indeed a range of intermediate possibilities. In a generalization of the mixing of exactly two points on the correlation envelope to form a mixing line, a range of origins may be responsible for generating the observed air mass (as postulated by *Plumb et al.* [2000]). In the example considered, the weighting function (“mixing kernel”) *h*(ζ) shown in Figure 3b characterizes such a situation whereby air from a range of low mixing ratios of ζ mixes with air from a range of high ζ. Hence *h*(ζ) dζ describes the relative fraction of an air mass originating in the infinitesimal region of reference tracer mixing ratio [ζ, ζ + dζ], so *h* constitutes a quantification of the air parcel's composition and hence mixing. Formally, *h* satisfies

with ψ (**x**, *t*) = (ψ_{1}, …, ψ_{p}) denoting the set of *p* different tracers modeled or observed at a given point of time and space, with **x** = (λ, ϕ, Θ) comprising the spatial coordinates longitude λ, latitude ϕ, and Θ. (In particular, Ψ describes the composition of the atmosphere at the isentropic level containing **x**.) Equation (1) expresses mathematically the notion that the composition of the atmosphere at (**x**, *t*) is the result of a linear mixing of points originating on the isentropic correlation envelopes. Correspondingly, the integral in equation (1) extends over the range of values which ζ attains on the isentropic surface of the measurement, restricted to one hemisphere. As a weighting function *h* is normalized and nonnegative, i.e., ∫ *h*(ζ; **x**, *t*) dζ = 1 and *h* ≥ 0. All tracers experience the same mixing. Hence if *h* is reconstructed from a set of tracers (as will be done in the following), the exact choice of tracers may be of little relevance. For simplicity, the additional dependencies of *h*, ψ_{i}, and Ψ_{i} on the spatial and temporal coordinates **x**, Θ, and *t* will be understood in the following.

[12] Under the assumption that equation (1) holds exactly in the real stratosphere for *p* different tracers, the mixing kernel *h* can be determined approximately. Let *H*(ζ; **x**, *t*) ≈ *h*(ζ; **x**, *t*) be an approximate solution to equation (1); *H* is taken to be a piecewise constant (i.e., a step-) function in ζ. It is hence denoted as *H*(ζ; **x**, *t*) = ∑_{k}*H*_{k}(**x**, *t*) *s*_{k}(ζ; Θ, *t*), where the *s*_{k} denote the rectangular functions describing the *k*th bin. The approximate mixing kernel *H* then does not exactly satisfy equation (1); instead, at every model grid point or for every measurement *H*(ζ) minimizes the quadratic form

under the conditions ∫ *H*(ζ) dζ = 1 and *H*(ζ) ≥ 0. An example is displayed in Figure 3c. Details about the particular choice of the norm ∥ ∥ in equation (2) measuring distances in *p*-dimensional tracer space, and how *H* is determined, are given in Appendix A. The computation of the correlation envelopes Ψ_{i} is explained in Appendix B. The result, in the cases considered here, may be expressed as a 5-dimensional vector field **h**(**x**, *t*) = (*H*_{0}, …, *H*_{4}) representing the step function *H*(ζ; **x**, *t*) which describe the contribution to the composition from any of five atmospheric regions. This choice of dimensionality means that with four atmospheric tracers available in the model simulation (described in section 3) and the normalization constraint the minimization problem is mathematically well posed. The vector field **h** is normalized such that ∑*H*_{k} = 1; the components *H*_{k} describe the fraction of air found at a given grid point originating from the *k*th interval (“bin”) [ζ_{k}, ζ_{k + 1}] of mixing ratios of the reference tracer ζ. Components of **h** will be displayed as isentropic maps.

[13] In Figure 3, if low mixing ratios of ζ denote core-vortex air and high mixing ratios subtropical air (as in Figure 2), then the considered air mass (the diamond in Figure 3a) has contributions both from the polar vortex and from the subtropics, indicating that it is of a mixed nature. In general, if the product of two bin amplitudes, *H*_{k} · *H*_{l}, does not vanish, this indicates contributions from both bins and hence mixing. However, note that due to the nature of the approximation, for an air parcel with ζ close to a bin boundary ζ_{k} the method will usually generate nonzero amplitudes of both *H*_{k − 1} and *H*_{k} without the parcel having to be of a mixed nature. Hence in the above it will only be of interest to analyze the products of nonneighboring bins with |*k* − l| ≥ 2, i.e., for long-range mixing in tracer space. The normalization implies that *H*_{k} · *H*_{l} ≤1/4. If the product equals 1/4, it means the product is composed of 50% contributions each from both bins.

[14] By its nature the above formalism does not account for diabatic motion. Hence in the formulation given here it cannot be used to infer long-term, “cumulative” effects of mixing, whereby mixing and diabatic motion change isentropic tracer-tracer correlation functions on similar timescales. In the following, we assess the method's capability of identifying regions of “instantaneous” mixing characterized by a relatively large distance between the vector of measured mixing ratios and their corresponding isentropic correlation function. A related, more general formalism that will allow to assess cumulative mixing is postponed to a future publication.

[15] In the following sections the method is applied to model and measurement data.

### 7. Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. The Method
- 3. The SLIMCAT Model
- 4. Meteorology of the Winter of 1999/2000
- 5. Results of Model Simulations
- 6. Application of Method to Measurements
- 7. Discussion
- 8. Summary
- Appendix A:: Details of the Inversion Procedure
- Appendix B:: Determination of Envelopes
- Appendix C:: Different Approximation to the Mixing Kernel
- Acknowledgments
- References

[38] The analysis presented above uses different tracers for the diagnosis of mixing in the ER-2 measurements, compared to the model. However, mixing is similar in both cases, indicating that the choice of tracers may be varied with little effect on the results, provided that the tracers span a range of sufficiently different photochemical behaviors so that their isentropic tracer-tracer correlations with a reference tracer are linearly independent. Mixing in the measurements is somewhat overdetermined, so individual tracers may be omitted from the analysis. This again leaves the results largely unchanged.

[39] In its present form the method requires the diagnostic determination of “correlation envelopes” that uniquely map one tracer onto another on an isentropic surface. In the SLIMCAT integration, if tropical data points are included, the data fall onto two distinct branches that make such a mapping impossible (Figure 1). An investigation of tropical measurements would be necessary to examine whether it is a model artefact; this is however beyond the scope of this paper. If it is not a model artefact, or indeed for this version of SLIMCAT, it limits the applicability of the method.

[40] Mixing is diagnosed in terms of a mixing kernel, which is approximated by the step function *H*. By the nature of the method details of the results depend on the position and number of bin boundaries ζ_{k} chosen to represent *H*. A variation of the bin boundaries results in a different distribution of amplitudes over the different bins, *H*_{k}. The spacing of bins is selected to cover the variability of the reference tracer N_{2}O in the region of interest, the polar vortex to midlatitudes. Given the amount of information available in the measured tracer data, an increase in the number of bins is mathematically possible but does not produce significant further insight. (For the model data, with no further assumptions an increase is not possible; see section 2.) Appendix C contains a sensitivity study in which *H* is chosen to be continuous and piecewise linear. The result supports the view that if a data point is some distance away from the isentropic correlation, the method will produce an overlap of distant bins, almost irrespectively of where the bin boundaries are placed and which approximation is chosen to represent *h*. In this respect the diagnosis of mixing is robust with respect to sensible variations of these parameters.

[41] A further assumption is that the signs of curvature of isentropic correlations do not change so that outer and inner sides of the curves may be defined. We argue that in the presence of a single transport barrier a change of the sign of curvature cannot occur because mixing on either side of the barrier straightens out the isentropic correlation while a single kink occurs at the position of the barrier. With a second transport barrier a change of the sign of curvature would be possible. In the cases considered here, changes in the sign of curvature of the isentropic correlations do not occur. The method does not explicitly require any particular sign of curvature of the tracer-tracer correlations. Here all correlations are positively curved, but the method would equally work with negative curvature, or indeed with negatively correlated tracer pairs.

[42] A fraction of mixing occurs locally in tracer space, i.e., it does not involve bringing air masses from very different parts of tracer space together. Instead, parcels neighboring in tracer space are perpetually mixed, adding to the compactness of isentropic correlations. *Plumb et al.* [2000] call this type of mixing “regular”. In the version presented here, due to insufficient numerical sensitivity, the formalism cannot capture regular mixing (section 2). This constitutes a limitation of the method. Sustained local mixing can lead to the formation of air parcels with more than two origins in tracer space, in which case the conventional “mixing line” approach leads to erroneous conclusions about the origins of such air parcels. In principle the method presented here would treat such situations correctly since it explicitly generalizes the mixing of exactly two points in tracer space, although in the present form a sufficient spacing of origins is necessary.

[43] Diabatic effects may change isentropic correlations in competition with mixing, and differential diabatic heating in midlatitudes may broaden isentropic correlations. The change of isentropic correlations due to subsidence in the vortex is accounted for by recomputing the correlation envelopes at every time mixing is calculated. Performing a trajectory study, *Sparling et al.* [1997] show that parcels that start out on a single isentropic surface experience substantial diabatic dispersion in the latitude −Θ plane. A determination of exactly how much this effect influences our results would require a separate study because of the different choice of analyses used here, the way diabatic motion is specified, and the fact that the diabatic dispersion obtained by *Sparling et al.* [1997] contains a reversible component due to meteorological dynamics.

[44] In addition to diabatic effects, chemistry changes isentropic tracer-tracer correlations. The lifetimes of tracers considered here range between 20 and 120 years (Table 2c). Even though equivalent stratospheric lifetimes, reflecting for example the average rates of photochemical loss at the considered altitudes, may be an order of magnitude smaller (e.g., *Minschwaner et al.* [1996] report a photochemical lifetime of 1 year for CFC-11 at 22 km in the tropics), they are not on the timescale of planetary waves responsible for mixing. Hence chemistry is thought to play a minor role here, but of course is essential in setting up the observed or modeled tracer-tracer correlations over longer periods of time. In some situations an inclusion of odd nitrogen or water vapor may be desirable but denitrification and dehydration would need to be accounted for carefully.

[45] As a note of caution, strictly the model analysis only shows that in the model filaments often have an anomalous composition indicative of mixing. However, in reality mixing presumably occurs at scales below those resolved by the models [e.g., *Morgenstern and Carver*, 1999, 2001, and references therein]. Hence an underlying assumption in transferring the results to the real atmosphere must be that the rate of filamentation in the models is similar to reality, because it is forced by the large-scale flow, and that filamentation is the rate-limiting step. In that case, the precise scale which diffusive mixing occurs at may be of little relevance as far as gross transport of long-lived species across the vortex boundary is concerned. For short-lived tracers, by contrast, it can have a substantial impact [*Thuburn and Tan*, 1997; *Searle et al.*, 1998]. A discussion of the real nature of mixing, involving an analysis of the fractal distribution of scales in stratospheric measurements [e. g., *Tuck and Hovde*, 1999], is beyond the scope of this paper.

[46] When applying the method to observations, the assumption is made that the correlation discerned from the measurements is indeed an isentropic correlation describing correlations in the region between the polar vortex and midlatitudes. A dependence of those functions on Θ cannot be discerned for points below 465 K but is clearly visible at higher isentropic levels. The assumption also depends on whether the measurements capture tracer-tracer variability on both sides of the polar vortex boundary. This appears to be the case, given the tight correlations of relatively reactive tracers like CFC-11 with N_{2}O (Figure 10). A change of tracer-tracer correlations between early and late flights of the campaign is not evident, suggesting that transport across the vortex boundary has not appreciably modified the composition of the vortex at the flight altitude in the intervening period. For the two flights analyzed here, an interleaving of vortex and extra-vortex air was found, similar to the findings of *Murphy et al.* [1989]. This supports the view that the boundary of the polar vortex is of a finite width and there is transport from the boundary of the polar vortex to midlatitudes and vice versa. The finding is largely in agreement with *Tuck* [1989], who suggests active exchange below 400 K and a suppression of isentropic exchange above this level over Antarctica, though the transition level may be at a higher potential temperature in the Arctic.

[47] The method will be modified in a forthcoming publication to assess cumulative mixing and possibly “mixing timescales” in the sense of *Minschwaner et al.* [1996].