## 1. Introduction

Large-scale transport and mixing has a profound impact on the global distribution and chemistry of trace constituents in the atmosphere. In the upper troposphere and lower stratosphere (UTLS), a combination of large-scale, quasi-isentropic stirring and the eddy-driven meridional overturning (Brewer 1949; Dobson 1956) is the main driver of global transport (Plumb 2002; Haynes 2005; Shepherd 2007). Climate models represent these processes reasonably well, but the estimates of transport vary significantly, depending on the numerical schemes, subgrid-scale parameterization, and resolution of the models (Hall et al. 1999; Rind et al. 2007). In particular, Hall et al. (1999) show that 3D models tend to overestimate the isentropic mixing in the stratosphere, rendering the mean age of air too young in the extratropics. Although it is easy to blame the shortcomings of the models for this discrepancy, one might also suspect that something about the atmospheric transport is intrinsically difficult to model.

One of the characteristics of atmospheric transport is its extreme inhomogeneity and anisotropy. Because of coherent structures in the flow, regions of fast stirring tend to be separated by semipermeable barriers to transport. The edge of the stratospheric polar vortex and the jet streams along the tropopause are among the examples of such barriers (Nakamura and Ma 1997; Haynes and Shuckburgh 2000a, b; Allen and Nakamura 2001). Jets hamper mixing in the cross-stream direction and concentrate the gradients of potential vorticity and of trace constituents. To the extent that the barriers act as the rate-limiting steps in the global transport, the disparate model estimates may be attributed, at least partially, to the difficulty in modeling the permeability of the barriers.

The goal of this paper is to quantify the sensitivity of the global transport and mixing to the properties of localized barriers. We will first develop a theory using a simple 1D model with a prescribed diffusion coefficient, with or without forcing. A barrier is represented in terms of a local minimum in the diffusion coefficient, and the rate of transport and the tracer structure are characterized as functions of the barrier geometry. We will then extend this analysis to the isentropic mixing in the UTLS region with the offline advection–diffusion calculations driven by the Met Office Stratospheric Analysis winds (Swinbank and O’Neill 1994). Using the effective diffusivity diagnosed in the equivalent latitude coordinate (Nakamura 1996; Haynes and Shuckburgh 2000a) and its relationship to the flux and the decay rate of the tracer, we will examine the effects of the spatiotemporal variations of the barrier properties on the global mixing.

The next section introduces the 1D model and addresses the role of a barrier under various model configurations. Section 3 describes the results of the isentropic transport calculations in the UTLS region. The final section summarizes the findings and discusses the implications for the global transport modeling.

## 2. 1D diffusion models of transport barriers

*q*on a bounded domain:Nakamura (1996) and Shuckburgh and Haynes (2003) show that 2D and 3D advection–diffusion–reaction equations can be reduced exactly to (2.1) if the area (volume) demarcated by the level set of

*q*is used as the coordinate

*y*. For example, for the isentropic transport in the stratosphere,

*y*is the equivalent latitude (Allen and Nakamura 2003) and

*Q*represents cross-isentropic transport and chemical sources and sinks. The effective diffusivity

*K*represents the net irreversible mixing driven by large-scale stirring. Note that

*K*is a function of

*y*and

*t*, and a transport barrier is characterized by a local minimum in

*K*(Nakamura 1996; Nakamura and Ma 1997; Haynes and Shuckburgh 2000a, b; Allen and Nakamura 2001; Shuckburgh and Haynes 2003; Marshall et al. 2006).

In this section, we prescribe *K* and *Q* and discuss the roles of a barrier on the global transport of *q*. For simplicity, we only consider time-independent *K* and *Q*. Some of the time-dependent effects will be discussed in the next section. To demonstrate the ubiquitous effects of a barrier, we will explore various configurations of the model—with or without forcing and different boundary conditions, etc.

### a. Unforced problem with fixed boundary values

*q*between two reservoirs. Although the boundary condition (2.2b) may be contrived as an example of the atmospheric tracer transport, this model serves a pedagogical purpose. When

*q*

_{0}≠

*q*

_{L}, Eqs. (2.2) permit a nontrivial steady-state solution:where

*F*

_{0}is a constant flux. The general solution to (2.2) is the sum of (2.3a) and a set of exponentially decaying modes; hence, (2.3a) dominates the solution at large

*t*. Because

*F*

_{0}is constant, |

*dq*/

*dy*| ∝

*K*

^{−1}. Thus, the tracer gradient is large where

*K*is small (i.e., the barrier region), consistent with the observed long-lived trace constituents (e.g., N

_{2}O and CH

_{4}) in the stratosphere whose gradients are concentrated at the edge of the polar vortex where effective diffusivity is small (Nakamura and Ma 1997; Strahan et al. 1994). As a concrete example, consider the piecewise constant profile of

*K*shown in Fig. 1a [defined by (A.1) in appendix A]. Inside the isolated barrier region of width

*d*, the diffusivity is smaller (

*K*

_{B}) than on the outside (

*K*

_{0}). The corresponding

*q*(

*y*) is piecewise linear, with the gradient being discontinuous and greater in the barrier region (Fig. 1b).

*K*and integrating in

*y*, one obtains an expression for

*F*

_{0}that does not involve

*y:*The angle brackets denote the average between

*y*= 0 and

*L*. Equation (2.3b) shows that, although the flux

*F*

_{0}is constant throughout the domain, its value does depend on the barrier geometry: for a given

*q*

_{0}and

*q*

_{L}, the flux is proportional to 〈

*K*

^{−1}〉

^{−1}or the harmonic mean of

*K*(denoted henceforth by

*K*

_{h}). For the profile of

*K*in Fig. 1a, it is readily shown thatFigure 2a plots this as a function of

*d*/

*L*(barrier width) and

*K*

_{B}/

*K*

_{0}(barrier depth). For a comparison, the arithmetic meanis shown in Fig. 2b. Although both show the same tendency—an increase with increasing

*K*

_{B}/

*K*

_{0}and decreasing

*d*/

*L*—it is clear that

*K*

_{h}(and hence the flux

*F*

_{0}) is much more sensitive to these parameters at

*d*/

*L*≪ 1,

*K*

_{B}/

*K*

_{0}≪ 1, namely, when the barrier is narrow and deep. Because

*K*

_{h}≤

*K*

_{a}(Schwarz’s inequality),

*K*

_{h}weights the small values in the barrier region more heavily. Furthermore,

*K*

_{h}is more sensitive to

*K*

_{B}than to

*K*

_{0}, which is demonstrated by the smallness of the ratioThis ratio, shown in Fig. 2c, is smaller than unity for most of the parameter space.

### b. Steady state with a prescribed forcing

*K*. Figure 3a plots |

*q*(0) −

*q*(

*L*)| as a function of barrier width and depth for the same diffusivity profile as in Fig. 1a and

*Q*=

*Q*

_{0}cos(

*πy*/

*L*). (Note the nonuniform contour intervals.) The cross-barrier variation increases sharply as the barrier deepens, although it is less sensitive to the barrier width. Figure 3b illustrates this with the profiles of

*q*for

*d*/

*L*= 0.1 and

*K*

_{B}/

*K*

_{0}= (1.0, 0.1, 0.05).

### c. Freely decaying modes with zero boundary fluxes

*y*= 0 to

*L*, one obtains, after some manipulation,where

*q̃*≡

*q*− 〈

*q*〉 is the tracer anomaly and

*q** is its rms-normalized value. Because the right-hand side of (2.10a) is negative, the variance of the tracer anomaly decays with time. (Note, however, that 〈

*q*〉 is time-independent.) A separation of variables

*q̃*≡

*q̂*(

*y*)

*e*

^{−αt}in (2.9) yields a Stürm–Liouville problem; hence, the solution is characterized by a linear combination of orthogonal normal modes. For the

*k*th eigenmode, (2.10) becomesThe asymptotic behavior of

*q̃*is dominated by the mode with the smallest decay rate

*α*(gravest mode). This minimum decay rate can be used as a rate of global mixing because the eigenmode is a global entity.

In (2.10a), *K*(∂*q**/∂*y*)^{2} is the (normalized) conditional dissipation rate (CDR), whose domain average is the decay rate of the tracer. By examining CDR as a function of *y*, one can determine the regions that contribute most to the tracer decay.

Figure 4 depicts the mode structures for the piecewise constant *K*. (The details of the solution procedures are found in appendix A.) When *K*(*y*) is symmetric about *y* = *L*/2, modes that are either symmetric or antisymmetric about *y* = *L*/2 are possible. In the left column of Fig. 4, we show the tracer structure (Fig. 4b), flux (Fig. 4c), and CDR (Fig. 4d) of the first antisymmetric mode for *d*/*L* = 0.1, *K*_{B}/*K*_{0} = 0.2 (Fig. 4a). On the right, similar profiles are shown for the first symmetric mode (Figs. 4f–h) with *d*/*L* = 0.4, *K*_{B}/*K*_{0} = 0.2 (Fig. 4e).

*q** for the antisymmetric mode (Fig. 4b) is similar to that in Fig. 3b: the gradients are single-signed but markedly enhanced in the barrier region. The flux structure (Fig. 4c) is modal and smooth. Notice that the flux is normalized by the rms value of

*q̃*,

*F** = −

*Kdq**/

*dy*, so it measures the efficiency of transport for the given amplitude of

*q̃*. It is interesting that the flux takes a maximal value at the center of the barrier, so it reveals no

*local*transport bottleneck. However,

*F** is reduced globally from the case without the barrier. The smoothness and the global response of the flux stem from the fact that only its second-order derivative is discontinuous (as long as

*q** is bounded and continuous), because (2.9a) may be arranged intoIn contrast, the gradient is discontinuous because

Because the CDR, *K*(*dq**/*dy*)^{2}, may be also written as −*F***dq**/*dy*, it is affected by the barrier both globally (through *F**) and locally (through *dq**/*dy*). Figure 4d shows a “spike” in the CDR in the barrier region, reflecting the enhanced gradient. The area under the CDR curve constitutes the decay rate of the mode [(2.11a)]. Although the spike is narrow, it contributes to about 60% of the decay rate. In contrast, the contribution from the same region is only 20% in the absence of the barrier. Thus, the barrier has a local effect of enhancing the relative contribution of this region to the tracer decay by modifying the CDR profile. On the other hand, its global effect is to decrease CDR on average (and hence decrease the decay rate), as indicated by the smaller area enclosed by the solid curve in Fig. 4d.

The first symmetric mode is characterized by a bulge in the barrier region, flanked by weaker gradients outside (Fig. 4f). For this mode, the flux changes sign at the center of the domain (Fig. 4g). The barrier reduces the magnitude of *F** globally, whereas the locations of the peak fluxes shift to the edge of the barrier region. Meanwhile, the greatest tracer gradients are found just inside the boundaries of the barrier; this is where the CDR attains peak values (Fig. 4h). The barrier region contributes to 80% of the decay rate, whereas in the absence of the barrier the contribution from the same region is only 31%. The symmetric mode is akin to the global distribution of long-lived trace constituents in the stratosphere such as N_{2}O and CH_{4}, whose mixing ratios maximize in the tropics where stirring is relatively weak (Nakamura and Ma 1997; Shuckburgh et al. 2001; Sparling 2000; Neu et al. 2003).

*d*/

*L*) and barrier depth (

*K*

_{B}/

*K*

_{0}). The decay rate of the first antisymmetric mode in Fig. 5a shows a distribution similar (though not identical) to Fig. 2a. This suggests that the harmonic-mean diffusivity

*K*

_{h}remains a useful first-order predictor of the decay rate of this mode. Indeed, if we assume that

*K*

^{−1}deviates only slightly from its mean 〈

*K*

^{−1}〉 and that the flux deviates only slightly from the first sine function sin(

*πy*/

*L*), then (2.12b) leads to the approximationA similar argument can be used to show thatis the decay rate of the first symmetric mode. The computed values in Fig. 5b are indeed approximately 4 times greater than those in Fig. 5a at large

*d*/

*L*, but the proportionality deteriorates appreciably at small

*d*/

*L*. A more accurate approximation may be developed formally based on a series expansion of

*K*

^{−1}(see appendix B). This leads to the following corrections to (2.14):where ε

_{1}and ε

_{2}are the coefficients of cos(2

*πy*/

*L*) and cos(4

*πy*/

*L*), respectively, in the Fourier cosine expansion of

*K*

^{−1}

*K*

_{h}. The approximate decay rates based on (2.15), shown in Figs. 5c,d, are virtually indistinguishable from the full solutions.

### d. Normal modes and linear relaxation

*q*

_{B}(

*y*) at a constant rate

*γ*.

*q*is self-adjoint,

*q*can be expressed in terms of a linear combination of orthogonal eigenmodes of the homogeneous (unforced) problem:Substituting in (2.16a), one obtainswhere

*α*

_{k}is the eigenvalue (decay rate) of the

*k*th eigenmode. The coefficient

*a*

_{k}may be determined from the orthogonality of the modes:Therefore, with knowledge of the eigenmodes and eigenvalues, the solution of (2.16) can be constructed from (2.19) and (2.17). The expression (2.19) shows that

*a*

_{k}in (2.17) depends on (i) the decay rate of the eigenmode relative to the relaxation rate and (ii) the projection of

*q*

_{B}on the eigenmode. For the same amount of projection, the mode with the least decay rate will attain the largest contribution to

*q*. A strong barrier can amplify this contribution further by lowering the decay rate

*α*

_{k}. On the other hand, for a large

*γ*(≫

*α*

_{k}), (2.19) asymptotes to

*q*=

*q*

_{B}.

Figure 6 illustrates the tracer structure for various *q*_{B}, *γ*, and *K*. In each column, the tracer is relaxed to the same *q*_{B} shown by the gray curves. It varies from left to right as cos(*πy*/*L*), 0.5[cos(*πy*/*L*) + cos(2*πy*/*L*)], and cos(2*πy*/*L*). The value of *γ* is the same within each row and it increases downward. In each frame, the tracer structure is shown for *K* given by (A.1) with *d*/*L* = 0.3 and (*K*_{0}, *K*_{B}) = (0.1, 0.01) (solid curve) and for a uniform *K* (*K*_{0} = *K*_{B} = 0.1; dashed curve). The overall tracer structure is dictated by *q*_{B}, but for small *γ*, diffusion manages to keep the tracer profile away from *q*_{B} (Figs. 6a–c). As expected, the tracer structure is well captured by the first antisymmetric and symmetric modes for the antisymmetric and symmetric forcing, respectively (Figs. 6a,c). When the forcing has an equal weight on the opposing symmetries (Fig. 6b), the tracer leans toward the antisymmetric mode because it has the smallest *α* [(2.14), (2.19)]. As *γ* increases, the tracer is more constrained to *q*_{B}, but the presence of the barrier still makes an unmistakable contrast in the tracer gradients (Figs. 6g–i). Notice that the tracer gradient can locally exceed the gradient of *q*_{B} in the presence of the barrier.

The annual-mean isentropic distribution of long-lived stratospheric trace constituents (N_{2}O, CH_{4}, CFCs, etc.) is roughly symmetric about the equator. This implies that the differential vertical advection associated with the upwelling in the tropics and downwelling in the extratropics has a large projection on the first symmetric mode, overcoming the faster decay rate compared to the first antisymmetric mode. Meanwhile, ozone in the upper stratosphere under the solstice condition is more antisymmetric about the equator because the forcing there is primarily photochemical and reflects the maximal (minimal) insolation at the summer (winter) pole.

## 3. Decay of tracer variance in the upper troposphere and lower stratosphere

The analyses in the foregoing section show that the asymptotic structure of the tracer and global mixing properties (fluxes and decay rates) depend sensitively on the geometry of the barrier and that the harmonic-mean effective diffusivity is a useful first-order predictor of these mixing properties. We will now test these ideas for the isentropic transport in the UTLS region by analyzing a freely decaying numerical tracer in the offline 2D advection–diffusion calculation driven by the Met Office Stratospheric Analysis winds (Swinbank and O’Neill 1994), using the equivalent latitude coordinate. For a complete description of the calculation, see Allen and Nakamura (2003) and Nakamura (2007). Briefly, on each isentropic surface, the tracer is initialized on 17 October 1991 as the sine of the latitude. Then the 2D advection–diffusion equation is integrated in time through 31 December 2005, using the nondivergent part of the isentropic winds interpolated vertically from the isobaric surfaces of the Met Office analysis. Where the isentropic surface goes “underground,” the surface winds are used to ensure global coverage. Each isentropic run is independent of the other levels. A spectral truncation of T115 and a constant diffusion coefficient of 70 588 m^{2} s^{−1} are used. (Truncation of the advecting winds is equivalent to T48.) A 14-yr long global tracer record is created with sampling 4 times daily. From this tracer output, we compute the 6-hourly effective diffusivity as a function of equivalent latitude using the method described in Nakamura and Ma (1997).

### a. Decay rates and effective diffusivity

The thick solid curves in Fig. 7 show the natural logarithm of the global-mean tracer variance on three different isentropic surfaces as functions of time. The variances decay approximately exponentially, and despite the seasonal and daily variations in the advecting winds, the decay rates are remarkably steady over the years. (There is a slight change in the decay rates after around 2000, which we will see more clearly in Fig. 12.) It is found that the tracer field maintains the monotonic pole-to-pole gradients throughout the integration at all levels, although the variance decays by many orders of magnitude. The preservation of the large-scale structure, together with the nearly constant exponential decay rates, strongly suggests the manifestation of an “eigenmode” (in an extended sense, because the flow is unsteady; see Pierrehumbert 1994; Sukhatme and Pierrehumbert 2002; Fereday et al. 2002 for related discussions).

*ϕ*

_{e}(Allen and Nakamura 2003), this equation is written as (Haynes and Shuckburgh 2000a; Marshall et al. 2006)where

*κ*is the numerical diffusion of the model and

*N*is the local Nusselt number, calculable from the instantaneous geometry of the tracer. This equation corresponds to (2.9a) of the previous section, except that effective diffusivity

*K*is also a function of time. To test the consistency of the diagnostic, we integrate (3.1a) in time with the initial condition

*q*(

*ϕ*

_{e}, 0) = sin

*ϕ*

_{e}, using the 6-hourly

*K*saved in the advection–diffusion calculation. The obtained tracer variance is shown in the dashed curves in Fig. 7. The results closely reproduce the full solution of the advection–diffusion at all levels, although the 1D calculations slightly overestimate the decay rates. The difference in the average decay rate is 2%–3%, and it is likely due to discretization errors and the effect of transients unresolved by the 6-hourly

*K*used to drive (3.1a).

The decay rates in Fig. 7 are nearly constant in time but they are distinct at the three levels. In view of the eigenanalysis in section 2c, we hypothesize that the decay rates depend on the geometry of *K*. Figure 8 shows * K* (the overbar denotes the climatological-mean annual mean) at the three isentropic levels. There are indeed considerable differences, in both magnitude and structure, among the three profiles. At 290 K,

*varies over a factor of 10 between the extratropics, where the isentrope resides in the storm tracks, and the tropics, where it samples mostly the surface air because the isentrope lies below surface (Fig. 8a). At 350 and 430 K, the range of variation is much smaller. At 350 K,*K

*has minima in the subtropics of both hemispheres, where the isentrope intersects with the subtropical jets (Fig. 8b), whereas the 430-K surface lies above the tropopause and the minimum*K

*occurs over the equator (Fig. 8c).*K

*α*

*α*

*e*-folding time of ∼145 days) are observed at 315 and 390 K, whereas a local minimum (∼190 days) is found at 335 K, and the decay rate decreases monotonically above 390 K. Figure 9a also shows the profiles of the approximate decay rates. The dashed curve is based on the approximationwhere

K

_{h}is the harmonic mean of

*(*K

*ϕ*

_{e}) and

*a*is the planetary radius. This corresponds to (2.14a) in the spherical coordinate. The solid curve represents a correction to (3.2a), analogous to (2.15a):where ε

_{2}is the coefficient of the second Legendre polynomial

*P*

_{2}in the Legendre expansion of

K

^{−1}

K

_{h}. For the derivation of (3.2), see appendix C. Although the details differ, 2

*α*

*α*

_{0}, and 2

*α*

_{1}all show broadly similar structures. Their interrelationship is shown in Figs. 9b,c as scatterplots. The 2

*α*

_{0}− 2

*α*

*α*

_{1}− 2

*α*

*α*

_{0}and 2

*α*

_{1}overestimate 2

*α*

It is found that 2*α*_{0} and 2*α*_{1} are more consistent with the decay rates of the gravest modes of the eigenvalue problem for the climatological-mean, annual-mean effective diffusivity. [To solve (3.1) as an eigenvalue problem, the domain is discretized with 180 equally spaced equivalent latitudes and the standard QR algorithm (Press et al. 2007) is used.] The variance corresponding to these eigenmodes is plotted in Fig. 7 with thin solid lines. The decay rates of the modes are 15%–20% greater than the time-mean decay rates of the full solution at all levels. This discrepancy is due to the temporal variation of *K* in the full problem and will be addressed shortly.

### b. Eigenmode and annual-mean tracer structure

Figure 10 compares the structure of the eigenmode with that of the time-mean tracer in the time-dependent initial-value problem. To prevent the exponential decay from affecting the time average, we first normalize the tracer at each time by its rms value [(2.10b)]. The time-mean tracer structure * q** is well reproduced by the eigenmode at all levels (Figs. 10a–c). The fluxes of the modes also match the shape of the time-mean fluxes of the initial-value problem (Figs. 10d–f), but the magnitudes are significantly greater. The CDR also shows good agreement (Figs. 10g–i), but again, the eigenmode exhibits systematically higher values than the time mean, as expected from the higher decay rate. Notice that the peaks of CDR coincide with the peaks in gradients and the troughs in

*(see Fig. 8 for the corresponding*K

*profiles). Consistent with Fig. 4d, this confirms the primary contribution of the barrier regions to the variance decay.*K

### c. Effects of time dependence

*captures, to a first approximation, the time-mean decay rates and structure of the tracer of the full problem. In reality,*K

*K*varies with time as indicated by the standard deviations of Fig. 8. The systematic bias in the decay rates and flux structure of the eigenmode stems from the time dependence of

*K*, which we shall discuss now. Let

*f*

_{m}and

*g*

_{m}be the magnitudes of the flux and gradient of the eigenmode, respectively, both normalized by the rms value of the tracer. (We drop the asterisk to avoid clutter.) They are related asSimilarly,where

*f*and

*g*are the magnitudes of the normalized flux and gradient in the full problem and the prime denotes the departure from the time mean. Now letwhich leads to, from (3.3) and (3.4),In the examples of Fig. 10, the first term on the right-hand side of (3.6) proves negligible. Thus,

*−*f

*f*

_{m}largely reflects

*−*f

*f*

_{m}< 0, implying

The negative correlation between effective diffusivity and gradient is confirmed in their annual cycle. Figure 11 shows the seasonal climatology of effective diffusivity (Fig. 11a), gradient (Fig. 11b), and flux (Fig. 11c) at 350 K. This isentrope intersects the tropopause in the subtropics, and strong barriers appear in the winter subtropics where the axis of the jet is located, although in the Southern Hemisphere there also appears a hint of a secondary minimum along the subpolar jet (Fig. 11a). (See Fig. 8b for * K*.) Effective diffusivity and gradient are highly correlated spatially as well as temporally, and this correlation is largely negative (Fig. 11b): a minimum in the former is accompanied by a maximum in the latter and vice versa, with little time lag between them. Because the eigenmode captures only the spatial correlation between them (there is no temporal fluctuation in the mode), it overestimates the diffusive flux.

Figure 11 also shows the annual cycle of the normalized flux (Fig. 11c). A large seasonal variation is found at subtropical latitudes. Unlike the gradient, the flux is maximal in summer shortly before diffusivity is maximized, and it is minimal in winter when diffusivity is minimized. However, the flux is not entirely in sync with diffusivity because it is a product of diffusivity and gradient and the latter two are antiphase. The peak values of diffusivity and flux in Fig. 11 are both higher in the Northern Hemisphere than in the Southern Hemisphere, presumably reflecting the stronger summer monsoon circulation in the former (Haynes and Shuckburgh 2000b). Figure 11d summarizes the phase relationship among diffusivity, gradient, and flux at *ϕ _{e}* = 25°S. (The quantities are normalized to a zero mean and unit variance.)

*). The ratio of these quantities can be written asA term-by-term evaluation of (3.7) reveals that the leading contributor next to 1 is*

_{m}*f*

_{m}, which is negative. The last term in (3.7) proves negligibly small. Terms involving Δ

*g*are also small except in the Northern Hemisphere extratropics, where they are positive and outweigh

*f*

_{m}. Thus, the above ratio is less than unity for the most part, consistent with Figs. 10g–i. Another, more direct way of showing the difference in the decay rates is based on the approximation

*α*∝

*K*

_{h}[(3.2), but we assume that the relation also holds in time]. If the temporal fluctuation of

*K*is sufficiently small, it can be shown explicitly (appendix D) that

K

_{h}≥

*K*will be greater than the time mean of fluctuating decay rate.

### d. Role of barriers in the shift of decay rates

In Fig. 7 we saw that the decay rates of the tracer variance were remarkably steady over many years. This may seem surprising given the highly variable advecting winds, but it attests to the regulatory effect of the transport barriers whose features are largely periodic in time (the annual cycle being the leading mode of oscillation). If *K* is time-periodic, (3.1a) in its discretized form can be reduced to a standard eigenvalue problem by Floquet’s theorem, so the asymptotic behavior remains exponential.

However, if the nature of the barrier changes over the years in an aperiodic fashion, the exponential behavior may be disrupted. The two curves in Fig. 12a reveal a systematic change in 2*α**α**α*_{0} and 2*α*_{1} as well (Figs. 12b,c), so the shift must be due to a change in *K* and ultimately in the advecting winds. Such change may not necessarily be of natural causes—indeed, in this case, there was a major change in the assimilation method in the Met Office Analysis implemented in November 2000 (Lorenc et al. 2000). A systematic change in the analysis winds is a likely cause of the shift in *K*. However, even if the shift is spurious, it is still important to recognize the role of the barriers in regulating its effect.

Figure 13 shows the climatologies of *K* and CDR over the two periods at 350 K. Figure 12b tells us that _{h} at this level increased from 1992–99 to 2001–05. In fact, Fig. 13a shows that *K* increased at all *ϕ*_{e} but mostly in the tropics and high latitudes, away from the barriers. The CDR also increased over time (Fig. 13b), consistent with the increased decay rate (Fig. 12a). However, this occurs mostly in the subtropics of the Southern Hemisphere, where *K* is small and increased only marginally. The much greater increase in *K* in the tropics and Northern Hemisphere seems to have little effect on the CDR and _{h} and hence on the decay rate. A small change in *K* in the barrier region has a disproportionately large effect. This sensitivity is consistent with Fig. 2c.

## 4. Summary and discussion

In this paper, we have investigated the effects of an isolated transport barrier, characterized as a local minimum in effective diffusivity (Nakamura 1996; Haynes and Shuckburgh 2000a, b; Shuckburgh and Haynes 2003), on the global mixing. An idealized 1D model was used to investigate the sensitivity of the tracer structure, flux, and decay rates to the prescribed geometry of the barrier, with or without forcing. For all the model configurations considered, the barrier is found to play a significant role, particularly when it is deep and narrow. The harmonic-mean effective diffusivity (*K*_{h}) is proposed as a useful first-order predictor of the rate of global transport.

The same 1D formalism was used to diagnose the freely decaying passive tracer in the isentropic advection–diffusion problem driven by the Met Office winds in the UTLS region. The asymptotic behavior of the tracer is an approximate exponential decay, but the decay rate varies with isentropic levels. Consistent with the 1D model, the decay rate is found to correlate strongly with *K*_{h}. Both the decay rate and the structure of the tracer are captured to a good approximation by the gravest eigenmode of the time-mean effective diffusivity. The CDR distribution reveals that the barrier region is the primary contributor to the tracer decay.

In the full time-dependent problem, the time-mean decay rate and flux of the tracer are 15%–20% less than those of the eigenmode of the time-mean diffusivity. This is because the temporal fluctuations of effective diffusivity and gradient are anticorrelated. We have also identified an appreciable (but probably spurious) shift in the decay rates after about year 2000. The change in *K*_{h} is consistent with this, and it primarily reflects the modest change in the diffusivity in the barrier region rather than the more sizeable changes in the stirred regions.

The sensitivity of the global atmospheric transport and mixing to the barrier properties poses a challenge to the modeler to the extent that the barrier properties are sensitive to the errors in the advecting winds and model numerics. As a demonstration, Fig. 14 shows the tracer variance as well as * K* and

^{2}s

^{−1}; the resolution of the model is the same). The decay rate is about 30% greater for the case with higher numerical diffusion (Fig. 14a). The profiles of

*for the two cases in Fig. 14b differ only modestly in magnitude, but the higher numerical diffusion significantly increases the minimum*K

*in the barrier region. This enhances*K

Marshall et al. (2006) show that effective diffusivity is insensitive to the choice of numerical diffusion as long as the Peclet and Nusselt numbers are large (see their Fig. 3). This condition is difficult to meet in the barrier region, where the Nusselt number is small by definition [(3.1)]. To achieve convergence in *K*, a sufficiently small numerical diffusion is necessary to ensure a large Nusselt number in the barrier region. However, this generally requires a higher model resolution and may not be a viable solution for climate simulations. Furthermore, if the Nusselt number is truly close to unity (i.e., if the large-scale stirring is negligible), then the permeability of the barrier is determined by the numerical diffusion. Then the rate of global transport will be dictated by the model’s numerical diffusion, explicit or otherwise, and therefore will become resolution-dependent. The presence of barriers may at least partially account for the disparate estimates of the mean age of air among the models as mentioned in the introduction (Hall et al. 1999). Hall et al. also show that most climate models and chemical transport models appear to overestimate isentropic mixing in the lower stratosphere, which suggests that the barriers are too diffusive in most models. It remains to be seen if one should seek convergence by increasing the model resolution and reducing the numerical diffusion (thereby increasing the Nusselt number), relying on the (weak but finite) large-scale stirring in the barrier region, or whether it is more fruitful to constrain the numerical diffusion of the barrier region from observation. Either way, a careful attention should be paid to the cross-barrier fluxes for future improvements in the transport modeling of the UTLS region.

## Acknowledgments

The author thanks two anonymous reviewers for their constructive criticisms on the first draft of this paper, which led to a substantial improvement. This work has been supported by NSF Grants ATM-0230903 and ATM-0750916. Views expressed herein are solely those of the author and do not necessarily reflect the views of the NSF.

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## APPENDIX A

### Piecewise Continuous Diffusion Model

*K*

_{B}over a finite width

*d*at the center of the domain, which represents an isolated transport barrier. By substitutingin (2.9), we obtainWe assume continuity in

*q̂*at the boundaries of the barrier

*y*= (

*L*±

*d*)/2. The symmetry of the problem about

*y*=

*L*/2 permits either the antisymmetric modeor the symmetric modeas possible solutions that satisfy (A.3). Here

*q*

_{0}is an arbitrary nonzero constant. Continuity of the flux −

*Kdq̂*/

*dy*at

*y*= (

*L*±

*d*)/2, together with (A.4) and (A.5), leads to the following dispersion relationships:

Both (A.6) and (A.7) admit multiple roots for the decay rate *α*, but we are primarily interested in the gravest modes (the modes with the smallest nonzero *α*). We obtain these by solving (A.6) and (A.7) with an iterative method. Figures 5a and 5b show *αL*^{2}/*K*_{0} of the gravest antisymmetric and symmetric modes, respectively, as functions of *d*/*L* (barrier width) and *K*_{B}/*K*_{0} (barrier depth).

## APPENDIX B

### Derivation of (2.15)

*K*. Without loss of generality, we can nondimensionalize as

*L*=

*π*and 〈

*K*

^{−1}〉 = 1. If

*K*

^{−1}(

*y*) ≡ 〈

*K*

^{−1}〉 = 1; then the first antisymmetric mode (which is symmetric in flux) isIn the dimensional form, (B.2) corresponds to (2.14a). Now we perturb

*K*

^{−1}aswhich assures 〈

*K*

^{−1}〉 = 1 and the symmetry in

*K*about

*y*=

*π*/2. The corresponding flux structure will bewhich satisfies (B.1) and the mode symmetry. Substituting (B.3) and (B.4) in (2.12b),We determine

*α*and

*δ*

_{m}by expanding (B.5) through

*O*(|ε

_{n}|) and

*O*(|

*δ*

_{m}|) terms and by requiring that the coefficient of each harmonic vanish. Usingthis yieldswhere (B.7) arises from the vanishing of sin

*y*terms and (B.8) from sin(2

*m*+ 1)

*y*terms. The dimensional form of (B.7) is (2.15a). From (B.7),

*α*> 1 for ε

_{1}> 0 (which makes

*K*greater at the center of the channel than at the boundaries), whereas

*α*< 1 for ε

_{1}< 0 (

*K*is smaller at the center). For the profile of

*K*given by (A.1), the Fourier cosine transform of

*K*

^{−1}givesWhen (B.9) is substituted in (B.7) and (B.8), the approximate eigenvalue and eigenmode are obtained. A similar argument may be used to derive the following approximations to the first symmetric mode:whereThe dimensional form of (B.10) is (2.15b). Figures 5c and 5d show the decay rate

*α*based on (2.15a), (2.15b), and (B.9). The results are nearly indistinguishable from the numerical solutions of (A.6) and (A.7) shown in Figs. 5a and 5b.

## APPENDIX C

### Derivation of (3.2)

*K*leads not to (2.12b) but towith the boundary conditionwhere

*μ*≡ sin

*ϕ*

_{e}. Analogous to (B.3) and (B.4), we set

*a*= 1 andwhere

*P*

_{n}(

*μ*) is the Legendre polynomial of order

*n*and |ε

_{n}|, |

*δ*

_{m}| ≪ 1; ε

_{n}=

*δ*

_{m}= 0 gives an exact solution to (C.1) and

*α*= 2, whose dimensional form is (3.2a). By substituting (C.3) and (C.4) in (C.1) and retaining the terms up to

*O*|ε

_{n}| and

*O*|

*δ*

_{m}| and then usingetc., one obtainsTaking the average between

*μ*= −1 and 1 and using 〈

*P*

_{m}〉 = 0 for

*m*≥ 1, (C.8) reduces toSince 〈

*P*

_{0}〉 = 1 and 〈

*P*

^{2}

_{2}〉, this yields

*α*= 2/(1 − 0.2ε

_{2}), whose dimensional form is (3.2b). To obtain ε

_{2}from

*K*, use the orthogonality of Legendre polynomials:

## APPENDIX D

### Proof of (3.8)

*K*is small:Because Schwarz’s inequality ensuresit suffices to showto prove (D.1). Applying Taylor expansion to

*K*

^{−1}under (D.2),Hence,Taking the time mean, (D.6) becomeswhich is (D.4).