a. Effective diffusivity

Effective diffusivity has been introduced by Nakamura (1996) and proceeds from a transformation of the advection–diffusion equation for a tracer with concentration c,

to tracer-based coordinates where the contour Γ(C) with c(x, y) = C is used as a coordinate line. Embedded contours are labeled by their area A_C. When the tracer has a mean pole to equator gradient, the area on the sphere is related to the equivalent latitude ϕ_e by A_C = 2πR²(1 − sinϕ_e), where R is the radius of the earth. Hence, C can be considered as a function of ϕ_e and t. The minimum length L_e(ϕ_e) of a tracer contour with area A_C(ϕ_e) is L_e = 2πR cosϕ_e, corresponding to the case in which the contour is aligned with a latitude circle.

It has been shown by Nakamura (1996) and Haynes and Shuckburgh (2000b) that, when the velocity u is nondivergent, (1) can be transformed to

where

It is assumed that ∇c does not vanish over the contour (this is generic if there are no saddle points). The value of L_eff is larger than the length L of the contour, but in practice the two quantities are fairly close (Shuckburgh and Haynes 2003; Scott et al. 2003). Both are generally much larger than L_e, because the contour is folded many times by chaotic advection, generating tracer filaments. The actual length results from a balance between continuous generation of small-scale filamentary structures and smoothing by dissipation. Here, dissipation κ is a simple uniform diffusion that is meant to represent subgrid-scale three-dimensional turbulence (Legras et al. 2005). The effective diffusivity takes into account the enhancement above the minimum length of the tracer contour length by the effects of stirring and mixing, and it is given from Eq. (2) by

The effective diffusivity is obtained daily by analyzing the contours of a chosen “probe” tracer evolving according to an advection–diffusion equation where advection is by the ERA-40 winds (see Haynes and Shuckburgh 2000a,b). The tracer evolution calculation is performed by a pseudospectral method using a T159 spherical harmonics representation and associated collocation grid. The tracer is initialized with a concentration proportional to the sine of the latitude. The parameter κ is a constant numerical diffusivity (taken as 1.5 × 10⁵ m² s⁻¹) used to parameterize the irreversible effect of small-scale turbulence. It has been shown that the effective diffusivity is largely independent² of the value of κ. A repeat calculation with a numerical diffusivity of 5 × 10⁵ m² s⁻¹ indicated little difference in the results. The equivalent contour length L_eff is calculated from the tracer each day, after a spin-up time of one month,³ as a function of the equivalent latitude ϕ_e with spacing of approximately 1°. From this, κ_eff is then obtained.

b. Finite-size Lyapunov exponents

In the absence of diffusion and sources, the evolution of an infinitesimal line element δx and of a passive tracer gradient ∇c follow very similar equations, because the variation δc = δx · ∇c is preserved by the motion. We have

where D/Dt means the time derivation along a given trajectory x(t), along which the motion is linearized. Over a time interval [t₀, t₀ + τ], a linear application 𝗠(x₀|t₀, t₀ + τ) is defined that maps a line element or a tracer from its initial value at x₀ = x(t₀) to its final value at x(t₀ + τ):

When computing δx(t₀ + τ) for finite time, one has the choice of prescribing the integration time τ or the ratio δx(t₀ + τ)/δx(t₀) between the initial and final separation. The first choice leads to the finite-time Lyapunov exponent calculation, whereas the second choice defines the finite-size calculation.

More precisely, FSLE is defined as (Ott 1993; Aurell et al. 1997)

where τ is the time at which the initial perturbation of size δ₀ has grown to δ, and the maximum is taken over the initial orientation of the line element or the tracer gradient. For small δ₀ and δ but large δ/δ₀ (or large τ, for the case of finite-time calculations) and if the flow is ergodic, λ[x(t₀), δ(t₀), δ] tends to a unique λ as a result of the Oseledec theorem (Oseledec 1968). In the other cases, λ exhibits large spatial and temporal variations, which represent the variations of stretching properties of the flow. In particular, the length of a contour Γ should grow exponentially with time, on average, for a length of time t (i.e., L = e^λ̂t, where λ̂ is the most probable value of λ in the domain spanned by the contour).

The calculation of Lyapunov exponents is well-defined forward and backward in time. As we shall see, both calculations contain valuable and complementary information on mixing.

The finite-size Lyapunov exponents are computed on a quasi-regular latitude–longitude grid, with a 0.5° spacing in latitude. Winds from the ERA-40 spectral T159 representation (as used for the effective diffusivity calculation) are interpolated in time using cubic splines and bilinearly in space from an intermediate projection to a 480 × 242 Gaussian grid. Trajectories of Lagrangian parcels are constructed both forward and backward in time. At each grid point, the trajectories starting at the four cardinal corners of a small square with a diagonal δ₀ = 0.1° are followed until a prescribed relative separation δ = 20δ₀ is reached for one of the diagonals. Other ratios of δ/δ₀ in the same order of magnitude (5, 10, and 50) have been also tested and provide very similar results. The finite-size Lyapunov exponents and vectors (which, in the following, will be referred to as Lyapunov exponents and Lyapunov vectors for simplicity; see appendix A for more details) are obtained from the singular values and directions of the matrix that transforms the initial square into a parallelepiped (Ott 1993; Koh and Legras 2002). Other details are in Mariotti et al. (1997) and Joseph and Legras (2002).

a. Relation between effective diffusivity and Lyapunov-derived quantities

In a turbulent system, a passive tracer is stirred in filamentary patches. In the early stage of their life cycle, filaments intensify local gradients by creating fingered intrusions between regions of different tracer concentrations. Such a gradient intensification mechanism is a precondition to mixing, because the redistribution of tracers in progressively thinner filaments eventually enhances the irreversible dispersion effect of small-scale turbulence.

The effective diffusivity and the Lyapunov exponent calculation provide two different approaches for detecting filamentation events. The effective diffusivity probes the velocity field with a passive tracer and a background diffusion. The enhanced dispersion effect resulting from the stirring is measured indirectly by averaging the filamentation events occurring along tracer contours. A contour grows during the initial stage of filamentation and retracts when its (thin) content is dispersed by the background diffusion. If the turbulent field has reached (or is close to) steady-state conditions, then growth and retraction events over an entire contour are at statistical equilibrium and the contour length evolves toward an asymptotic value. Longer contours correspond to a stronger filamentation process and to a more effective enhancement of the background diffusion. Therefore, by quantifying the contour length, an estimate of mixing is obtained in terms of an effective diffusion. However, the contour averaging nature of the effective diffusivity is the main limitation of this technique, because mixing variability along an individual contour cannot be resolved.

Lyapunov exponents and Lyapunov vectors detect local stretching properties of a velocity field with both longitudinal and latitudinal variability. In the case of filamentation events, forward Lyapunov calculations provide intensity and future direction of tracer gradient intensification, whereas backward calculations estimate the present orientation of a tracer gradient that has been passively advected. Strong filamentation is typically associated with large Lyapunov exponents because of the presence of strong chaotic regions. On the contrary, weakly mixing structures, such as the cores of slowly evolving vortices, are often associated with stagnation, low particle separation, and therefore low values of Lyapunov exponents (Joseph and Legras 2002; see also Garny et al. 2007).

Hence, one might expect the effective diffusivity to be large when the Lyapunov exponent is large. In fact, a relationship can be derived from the following heuristic model for blob dilution (Shuckburgh and Haynes 2003). Consider a finite blob of fluid in a domain of size L₀ × L₀. As time τ goes on, under the repeated action of stretching and folding, the fate of the blob is to invade the whole domain. It does so by elongating as e^λ̂τ, and its lateral size is bounded by L_d. When the blob fills the domain, its length L is such that LL_d = L₀² (i.e., L² = λ̂L₀⁴/κ); hence, the effective diffusivity is κ_eff ≈ λ̂L₀².

The actual calculation of effective diffusivity involves replenishing of the tracer by a large-scale gradient but follows a similar scaling⁴ if the contour fills a significant portion of the accessible domain.

b. Particle dispersion and mixing in a shear-dominated flow

There are, however, cases in which particle separation does not always act together with folding, notably in shear-dominated regions within jet cores. In this case, the use of the Lyapunov exponent as a diagnostic of mixing can be misleading.

In a chaotic region, the stirring of a passive tracer is always active and eventually leads to a uniform tracer concentration. In contrast, the asymptotic effect of a pure shear is not the suppression of the tracer gradient but its orientation parallel to the shear direction (Fig. 1). Calling ϕ the angle between the gradient direction and the shear direction, calling Λ the shear intensity, and solving the advection equation for the case of a linear tracer gradient, one finds

Once the angle between the tracer gradient and the shear direction is zero, the tracer is in a stationary condition. No mixing occurs at this stage, and this mechanism maintains asymptotically a large-scale tracer gradient. In this sense, shear is more consistent with a transport barrier than a mixing region, even if dissipation of the tracer gradient can occur during the transient or in response to perturbations (e.g., induced by shear instabilities).

On the other side, particles that do not strictly align on a streakline are separated by the shear. There is no mixing associated with this separation if the particles carry the same value of tracer and are thus indistinguishable. In a pure shear flow, the separation is linear and leads asymptotically to a zero Lyapunov exponent. However, over finite time this separation can be large. Moreover, when small random strain with amplitude γ is added to the shear, it is shown in appendix B that the Lyapunov exponent no longer vanishes but scales as Λ^2/3γ^1/3, which is dominated by the large value of the shear. For this reason, Lyapunov exponents may exhibit large values, and a maximum in Lyapunov exponents may appear where the effective diffusivity has a minimum. Examples will be shown later, and other cases of failed relation between effective diffusivity and Lyapunov exponent of this type can be found in simple analytic flows (Shuckburgh and Haynes 2003).

c. Particle dispersion from a dynamical systems perspective

The ambiguous interpretation of large values of Lyapunov exponents and mixing is also apparent by looking at filaments from a dynamical systems perspective. By identifying the physical space with a phase space, tracer filament formation has been shown to correspond to the folding of the stable and unstable manifolds of hyperbolic structures that are embedded in the velocity fields. This folding process is typically associated with the presence of homoclinic and heteroclinic intersections of the manifolds of hyperbolic points (see, e.g., Beige et al. 1991; Wiggins 2005; Haller 2000; Lapeyre et al. 2001; Mancho et al. 2004; Shadden et al. 2005). These manifolds are often used as templates of transport properties. In fact, under general conditions, it can be shown that a passive tracer tends to align its gradient orthogonally to the unstable manifolds. When the manifolds fold in thin lobes, as in the case of a perturbed homoclinic and heteroclinic intersection, passive tracers are redistributed in filament patches, enhancing the background diffusion and eventually leading to mixing. Manifolds of hyperbolic points have typically larger stretching rates than the ones found in nearby regions. On the basis of this heuristic assumption, these manifolds are often unveiled as local maxima (ridges) in a map of Lyapunov exponents.⁵ Although the Lyapunov calculation also provides estimates of manifold orientation through the Lyapunov vectors, the Lyapunov exponent depends entirely on the stretching, not on the geometry of the folding. This complicates the use of the exponent alone for mixing, because the formation of longer and thinner filament is not necessarily associated with larger Lyapunov exponents. The problem can be exemplified by considering the two cases of unperturbed and perturbed homoclinic structures (see, e.g., Beige et al. 1991, their Fig. 1a; Joseph and Legras 2002, their Fig. 1). A tracer released in the vicinity of these structures tends to orientate its gradients orthogonally to the manifolds (with a relaxation time given by the Lyapunov exponent). These dynamics result in two very different tracer redistributions: transport by lobes and formation of longer interface surfaces (with enhanced mixing in presence of background diffusion) for the perturbed case and segregation for the unperturbed case. However, the values of Lyapunov exponents in the two cases do not have to be different. Similarly, the values of the Lyapunov exponents along a manifold are not necessarily larger where the manifold folds and mixing events occur than when the manifolds intersect at low angles, and a tracer gradient can be maintained for longer times. These observations suggest that the Lyapunov exponent, in order to correctly detect mixing events, needs to be complemented by some additional information (the Lyapunov vectors) for describing the intersection of stable and unstable manifolds. In the following, we will see how to obtain an analytical relation that combines Lyapunov exponents and vectors. This will be done by comparing climatologies of Lyapunov exponents to the effective diffusivity and by studying in more detail the case of shear flows that corresponds to the case in which stable and unstable manifolds intersect at low angles.

d. Modified Lyapunov exponents

If we assume that the effective diffusivity provides a good parameterization of mixing, then the fact that stretching alone is not always a correct proxy of mixing is apparent in Fig. 2a, which shows no compact relation between monthly means of effective diffusivity and monthly means of Lyapunov exponents averaged around equivalent latitude contours. From Figs. 3 and 4, one can see that the Lyapunov exponents (dashed line) and effective diffusivity (black solid line) are strongly anticorrelated in the vicinity of the subtropical jets, where transport is dominated by shear (with κ_eff indicating a minimum, but λ indicating a maximum). An examination of gridded maps of Lyapunov exponents (Fig. 5) also indicates that a stretching maxima appears to be located in correspondence with the strongest zonal segment of the subtropical jet. This confirms that the Lyapunov exponent alone can be a misleading diagnostic of mixing in shear-dominated regions.

However, the Lyapunov calculation also contains Lyapunov vectors that provide information on the geometry of the stretching that a passive tracer undergoes.⁶ As noted earlier, the eigenvector associated with the principal Lyapunov exponent of the backward calculation is known to indicate the orientation of the gradient of a passively advected tracer. Conversely, the eigenvector obtained for the Lyapunov calculation forward in time is the stable direction relative to future gradient amplification. A simple condition for the presence of shear is therefore given by the angle α between these two Lyapunov vectors. This is because in shear-dominated regions and large times these two directions tend to coincide, whereas they differ in the case of mixing regions (see appendix A for more details). As a first step toward a Lyapunov diffusivity, we propose a quantity that depends on the particle dispersion in the future (the forward Lyapunov exponent λ) and also on the angle α between forward and backward Lyapunov vectors, so that the values of strong Lyapunov exponent in shear regions are suppressed:

The factor sin²α modulates the growth rates of the tracer gradient [see Eq. (A4)]. It can be justified analytically considering the dynamics of a fluid parcel in a shear-dominated flow with random perturbations, as discussed in appendix B. The angle α increases when the thinning of filaments develops. The multiplication by the factor sin²α thus emphasizes the Lyapunov values where the filaments are thinner and suppresses the values where shear dominates; hence, it is expected to have a better correlation with the effective diffusivity.

Returning to Figs. 3 and 4, the modified Lyapunov exponent λ̃ (gray line) correctly represents a mixing minimum corresponding with the wintertime subtropical jet (although there are still differences with respect to the effective diffusivity; in particular, the weaker minimum in correspondence with the summertime subtropical jet is not well resolved). An examination of gridded maps of this quantity (Fig. 5) indicates that the inversion of the subtropical extrema is collocated with the most zonally active part of the subtropical jet (cf. the minimum of λ̃ and the maximum of λ over the west Pacific). Figure 2b shows that there is a more compact relation in the extratropics (red dots) between the monthly mean modified Lyapunov exponent and effective diffusivity compared to the unmodified version in Fig. 2a. In the extratropics, the correlation coefficient is r = 0.73. In the tropics (blue dots), the correlation is still weak (see later comments).

e. The Lyapunov diffusivity

Supported by the improved correlation between λ̃ and the effective diffusivity as well as by theoretical arguments (see appendix B), we define a Lyapunov diffusivity D_λ(θ, ϕ) as a quantity with the units of diffusion and a linear dependence on λ̃:

where κ_a and b are proportionality coefficients and L_eddy is a length scale representing the most energetic perturbations. As a first approximation, outside the tropics, one might expect L_eddy to scale as the Rossby deformation radius. However, there are times (e.g., when the flow is dominated by the Asian monsoon anticyclone) when this may be a very poor approximation. In light of this, we set L_eddy to a generic value of 1000 km, reducing at high latitudes (above 70°) as sine of latitude (i.e., with the deformation radius). We propose to find the coefficients κ_a and b by taking the average around equivalent latitude contours of Eq. (11) and fitting it to the effective diffusivity κ_eff. Figure 6 shows the temporal evolution of averaged around equivalent latitude contours, along with κ_eff. A similar temporal evolution can be seen in both quantities, reflecting the good correlation noted earlier.

When evaluating the coefficients in Eq. (11), we compensate for the fact that we are not properly representing the latitudinal variations in L_eddy by determining the values of κ_a and b separately for each equivalent latitude or in other words by allowing these values to represent the unaccounted variation in latitude. Specifically, we take the monthly means of averaged around equivalent latitude contours and the monthly means of κ_eff for the entire time period; at each equivalent latitude, we use a linear fit to obtain κ_a(ϕ_e) and b(ϕ_e). It should be noted that in the tropics, where divergent motion forced by (in the analyzed wind field, parameterized) convection is important, the length scale likely gets very small, possibly to that of the grid size, and the Lyapunov exponent may not be very meaningful. In addition, the weak temporal variability of both and κ_eff in the tropics (see Fig. 6) makes the fitting of Eq. (11) unreliable.

Figure 7 presents the results of the linear fit procedure. At each equivalent latitude outside the tropics (±15°), there is a good fit between and κ_eff, with a correlation coefficient always greater than 0.4. For these extratropical equivalent latitudes, the multiplicative coefficient b is typically ∼0.5 and the additive coefficient κ_a is typically small. The exception is in the Northern Hemisphere subtropics where, as previously noted, the scaling we have chosen for L_eddy may be erroneous because of the dominance of the Asian monsoon anticyclone. The correlation between and κ_eff is weak in the tropics; as previously discussed, the Lyapunov calculation may have less relevance there. Correspondingly, the D_λ values are dominated by the additive coefficient κ_a in this region, and the contribution from is small.

Figure 8 presents the results of the fit (gray) to the effective diffusivity data (black) for each season averaged over the period 1980–2001. It can be seen that, at all times, the linear fit reproduces the effective diffusivity well.

f. A climatology of local mixing at 350 K

The effective diffusivity for the UTLS region has been analyzed previously by Haynes and Shuckburgh (2000b) and Scott et al. (2003). Here, we focus on the ability of the Lyapunov diffusivity to detect longitudinal variability of isentropic mixing. Figure 9 presents the Lyapunov diffusivity as a gridded map that is time averaged for December 2000–February 2001 and as a climatology for the entire period 1981–2001.

The picture for December 2000–February 2001 demonstrates that there is considerable longitudinal variability in the mixing that arises from the longitudinal variability in the Lyapunov exponents (Fig. 5). At all latitudes, local mixing regions may be characterized by intensities several times stronger than their zonal mean.

The climatological picture highlights some interesting systematic features of the longitudinal variability. The effective diffusivity detects reduced mixing, indicating a barrier to eddy transport at the latitudes of the subtropical jets. The Lyapunov diffusivity uncovers a more complex picture, with sections of mixing minima separated by synoptic-scale regions of vigorous mixing along the subtropical jets.

In the Northern Hemisphere, the jet velocities are strongest in a climatological sense in the western Pacific and North Africa–Middle East and the barrier strength is also strongest in these regions. The strongest mixing in the region of the subtropical jet (2 × 10⁶ m² s⁻¹ in the climatological mean) is at the jet exits in the central and eastern sides of the ocean basins, extending into North America and northern Europe, respectively. These regions coincide with regions of enhanced Rossby wave breaking (Postel and Hitchman 1999). Poleward of the subtropical jet is a region of rather homogeneous mixing at all longitudes. In the Southern Hemisphere, mixing appears more uniform and relatively weaker than in the Northern Hemisphere in the climatological mean. A strong subtropical barrier is observed at all longitudes (strongest over Australia). There is also evidence of a barrier associated with the polar jet at 60°S over the Indian and Atlantic Oceans. Regions of enhanced mixing in the Pacific subtropics and at all longitudes along the coast of Antarctica roughly correspond to regions of enhanced Rossby wave breaking (Postel and Hitchman 1999; Berrisford et al. 2007). All these longitudinal variations in mixing will be examined in more detail in Part II.