Diagnosing transport and mixing using a tracer-based coordinate system

PHYSICS OF FLUIDS

VOLUME 15, NUMBER 11

NOVEMBER 2003

Diagnosing transport and mixing using a tracer-based coordinate system Emily Shuckburgha)

Laboratoire de Me´te´orologie Dynamique, E´cole Normale Supe´rieure, Paris 75005, France

Peter Haynesb) Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, United Kingdom

共Received 29 April 2003; accepted 21 July 2003; published 17 September 2003兲 The advection-diffusion equation for the concentration of a tracer may be transformed into a pure diffusion equation by using the area inside concentration contours as a coordinate. The corresponding effective diffusivity depends on the geometry of the tracer field, which is determined by the underlying flow. Recent studies have used effective diffusivity, calculated from a suitable tracer, as a qualitative indicator of the transport and mixing properties of a given flow. Here we show that the effective diffusivity may further be used as a quantitative diagnostic of transport and mixing. We use a family of incompressible two-dimensional time-periodic flows as a test-bench and compare the calculated effective diffusivity with other diagnostics. The results demonstrate how the effective diffusivity accurately captures the location and character of barrier and mixing regions. We also show that the effective diffusivity parameterizes the transport of particles relative to the tracer-based coordinates. These results support the use of effective diffusivity as a quantifier of transport and mixing, and thereby strengthen the conclusions of previous qualitative studies. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1610471兴

I. INTRODUCTION

between different mixing regions, one can regard the mixing regions as being separated by semipermeable or ‘‘partial barriers.’’ A partial barrier is ‘‘strong’’ if the time scale for transport across it is large, and ‘‘weak’’ if this time scale is small. Identifying a perfect barrier, particularly in a timeperiodic flow where Poincare´ sections are a useful analysis tool, is relatively straightforward. Identifying a partial barrier and quantifying the rate of transport across it is much less straightforward, but is of key importance in many problems in geophysics, plasma physics and chemical engineering where flows are usually aperiodic in time. For example, it is important to identify the position and strength of: Barriers in plasmas1 共since they alter the fusion properties兲; the barrier at the edge of the stratospheric polar vortex2 共since it is within this vortex that the greatest ozone depletion occurs each spring兲; and the Gulf Stream barrier3–5 共since it determines the distribution of geochemical properties兲. Quantifying the strength of mixing, measured by the rate of stretching of material curves, and its spatial variation in time aperiodic flows is also nontrivial and important in its own right, for example, in flows with reacting chemical or biological species 共relevant in geophysical situations as well as in chemical engineering processes兲. Up to now we have spoken of the transport and mixing properties of a flow. But a further key aspect of the transport and mixing problem is to predict the evolution of a passive scalar that is both advected by the flow and subject to molecular diffusion. 共We shall refer to such scalars as ‘‘tracers.’’兲 The action of diffusion is enhanced by the flow through differential advection which, by stretching and folding of material lines, produces small scale structures in the tracer field. One recent approach to quantifying different as-

The goal of this paper is to analyze carefully a heuristic, but apparently robust, method to quantify the transport and mixing properties of flows that have some coherent spatial organization. In common with much previous literature on this subject, here we use the term ‘‘mixing’’ to denote the stretching and deformation of material lines by advection that will inevitably, in the presence of molecular diffusion, lead to real molecular mixing. This means it makes sense to talk about the ‘‘mixing properties’’ of a flow, without needing to know, for example, the precise value of molecular diffusivity. A much-studied special class of flows with coherent spatial organization is kinematic chaotic advection flows arising from imposed time-periodic velocity fields. These manifest well-known features of mixing regions, in which particle trajectories are chaotic, separated by transportbarrier regions, in which many particle trajectories are integrable. These transport barriers are ‘‘perfect,’’ in the sense that a fluid particle advected by the flow that starts on one side of the barrier never crosses to the other side. A similar mixing region/barrier region structure is observed in many flows with more general time-dependence, including dynamically consistent geophysical flows, in which the velocity field obeys dynamical equations 共as opposed to kinematic flows where the velocity field is imposed兲. In these cases the barriers are unlikely to be perfect; nevertheless, if there is sufficient contrast in scale between, say, the time for transport within each mixing region versus the time for transport a兲

Electronic mail: [email protected] Electronic mail: [email protected]

b兲

1070-6631/2003/15(11)/3342/16/$20.00

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© 2003 American Institute of Physics

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Phys. Fluids, Vol. 15, No. 11, November 2003

pects of transport and mixing of a tracer has been to define a new quasi-Lagrangian coordinate system based on the concentration contours 共or surfaces兲 of the tracer. This has the advantage of separating the reversible effects of advection from the irreversible effects of advection and diffusion acting together. When the advection-diffusion equation is transformed into the tracer-based coordinate system, the reversible effects of advection are incorporated within the coordinate system itself and one obtains a diffusion-only equation with an ‘‘effective diffusivity’’ which quantifies the irreversible processes.6,7 In one sense effective diffusivity is simply a diagnostic of the diffusive flux of tracer across tracer contours. But the fact that the diffusive flux arises from the combined effect of advection and diffusion means that the effective diffusivity maps out some of the transport and mixing properties of the flow. In this spirit, recent papers have analyzed the transport and mixing properties of atmospheric flows, defined by velocity fields stored in meteorological datasets or generated by numerical models, by simulating numerically the advectivediffusive evolution of an artificial reference tracer and then calculating the effective diffusivity directly from the evolving tracer field.8 –10 In a few specific examples 共all from the winter polar stratosphere兲, it has been noted that the spatial variation of the effective diffusivity matches in a qualitative sense the picture given by other indicators of the transport and mixing properties, such as distributions of chemical species11 and measures of particle dispersion.8,12 Furthermore, it has been shown by Nakamura6 in the case of a nonlinear Kelvin–Helmholtz instability that the effective diffusivity is largest at the center of the shear layer, where there is known to be strong mixing, and smallest at the edges of the shear layer, where mixing is known to be weak. To date, however, there has not been any systematic, quantitative study assessing the robustness of effective diffusivity and comparing it with other measures of transport and mixing. Potentially, effective diffusivity is a diagnostic of transport and mixing that can be applied in many other contexts, beyond atmospheric flows, and equally well to time-periodic or to time-aperiodic flows. The derivation of the diffusion equation requires no approximation, and the effective diffusivity that arises is quite distinct from that which has been used widely to quantify the dispersion of particles or transport of tracers on large spatial and temporal scales, e.g., in spatially–temporally periodic or random flows 共e.g., see Sec. II of the review by Majda and Kramer13兲. Calculating the transport and mixing properties of a flow by solving the advection-diffusion equation for a tracer rather than by direct analysis of particle trajectories, might seem unnecessarily complicated and expensive. However, the fact is that identification of partial barriers is difficult using particle or trajectory based methods—it is easy enough to calculate the transport across a specified control surface, but identification of a partial barrier requires identification of a control surface, probably a moving control surface, across which the transport is minimized. An advected tracer naturally maps out the transport and mixing structure of the flow, and therefore, is a useful intermediary which solves the problem of identification of partial barriers. As we will demonstrate in this paper,

Diagnosing transport and mixing

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calculating the effective diffusivity is a systematic and objective way of extracting quantitative information about the transport and mixing properties of a flow from the tracer field. Consequently, this approach may be used to extract information about the transport and mixing properties of a time-aperiodic flow that could not easily be obtained using other techniques based on particle advection. At present, there is no precise mathematical theory relating effective diffusivity to transport and mixing properties of a flow, only heuristic arguments. Given this lack of a mathematical theory, the best approach to improving understanding seems to be careful numerical experimentation under controlled conditions. To that end, we use a family of timeperiodic flows as a test-bench to analyze further the properties of effective diffusivity. We adopt this approach not because these time-periodic flows are of great interest in themselves, but so we can exploit the large body of previous work on transport and mixing in such flows in making the comparison with effective diffusivity. We anticipate that lessons learned from this family of time-periodic flows will help strengthen our understanding of the effective diffusivity in the context of flows with more general time dependence. In Sec. II we present a straightforward derivation of the effective diffusivity equation. In Sec. III we apply the effective diffusivity diagnostic to a simple time-periodic chaotic advection flow with mixing and barrier regions. First we investigate whether the effective diffusivity, calculated from a single tracer field, is a well-defined, robust diagnostic of the flow. Then we evaluate whether the diagnostic can correctly identify the transport and mixing properties of the flow and we assess the ability of the effective diffusivity to quantify particle transport. Finally we compare the effective diffusivity to finite-time Lyapunov exponents 共as another commonly used diagnostic of transport and mixing兲. In Sec. IV we discuss the results and indicate avenues for further research. II. A DIAGNOSTIC OF TRANSPORT AND MIXING A. Tracer coordinate system

In this paper we investigate the transport and mixing properties of a flow by considering the behavior of a ‘‘reference’’ tracer, which could be a chemical species, temperature, density or potential vorticity. We use the level sets of the tracer 共contours in two-dimensional domains or surfaces in three-dimensional domains兲, labeled by the concentration value C, to define a new, curvilinear coordinate system;14 see Fig. 1. For each value C of the concentration, between the minimum C min and maximum C max , one may define the function D(C,t) as the area A or volume V of the region for which the tracer concentration value c(x,t) is less than or equal to C 关i.e., A⫽ 兰 H(C⫺c(x,t))dA and V⫽ 兰 H(C ⫺c(x,t))dV, where H is the Heaviside function兴 and ␥ (C,t) as the bounding contour or surface. We will demonstrate the usefulness of this formulation in the following sections, but first we will note some properties of the coordinate system. As illustrated in the Fig. 1共b兲, the region for which c(x,t)⭐C might not be simply connected, in which case contributions to D will come from each of the disconnected

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Phys. Fluids, Vol. 15, No. 11, November 2003

E. Shuckburgh and P. Haynes

FIG. 1. A schematic demonstrating the relation between a tracer field at a particular time, the volume 共a兲 or area 共b兲 within a tracer surface or contour, and equivalent height or latitude. Tracer contours with values C⫽C 1 , C 2 are indicated with C min⭐C1⬍C2⭐Cmax , as are the corresponding equivalent heights z e⫽z 1 ,z 2 and latitudes ␾ e⫽ ␾ 1 , ␾ 2 . 共Note that the tracer contour value corresponding to a fixed volume or area is time dependent.兲

parts of the region. D is a monotonic function of C, ranging from D(C min ,t)⫽0 to D(C max ,t)⫽D 共where D is the total area or volume of the domain兲, and hence it has a unique ˜ ,t),t) inverse function which is C(D,t) such that C(D(C ˜ ⫽C 关giving C(0,t)⫽C min(t) and C(D,t)⫽C max(t)]. The new curvilinear coordinates are C and s 共where s are the distance coordinates on the isotracer contour or surface兲 and the corresponding coordinate vectors are parallel and perpendicular to ⵜc. The Jacobian for the transformation from Cartesians to (C,s) is given by 兩 ⵜc 兩 共since in two dimensions the area element dA⫽dsdC/ 兩 ⵜc 兩 and in three dimensions the volume element dV⫽ds 1 ds 2 dC/ 兩 ⵜc 兩 ). It is useful to note that in two dimensions

⳵ ⳵C

冕

⳵ 共 • 兲 dA⫽ ⳵C c⭐C

冕

dsdC * ⫽ 共•兲 兩 ⵜc 兩 c⭐C

冕

ds , 共•兲 兩 ⵜc 兩 ␥

共1兲

where the second equality uses the fundamental theorem of calculus noting that the boundary of the region of integration corresponds to C * ⫽C, and an analogous result holds in three dimensions. The new coordinates may be used to define averaged quantities. A natural average of a second tracer with concentration ␹ (x,t) around ␥ (C,t), denoted 具 ␹ 典 A , is given by14

⳵ ⳵A

冕

c⭐C(A,t)

␹ dA⫽

冉冕 冊 冕 ds 兩 ⵜc 兩 ␥

⫺1

␥

␹

ds , 兩 ⵜc 兩

共2兲

where the equality follows from Eq. 共1兲. The procedure to calculate this average numerically is to tabulate the area integral for a set of values of A and then to estimate the derivative with respect to A. In three dimensions, 具 ␹ 典 V ⫽ ⳵ / ⳵ V 兰 c⭐C(V,t) ␹ dV. The above formulation is most useful when the flow field has coherent spatial structure so that the field of an advected, diffused tracer naturally maps out the structure of the flow. Where possible, depending on the geometry of the problem, it is convenient to define a new coordinate based on 14

D, but with dimensions of length, since such coordinates can aid physical intuition. For example, in two-dimensional flow in a periodic channel of length l, where the flow structure is such that tracers show, e.g., strong variation in the crosschannel direction but weak variation along the channel, an equivalent cross-channel coordinate y e may be defined by A⫽ly e . C may therefore be regarded as a function of y e and t, rather than of A and t. On a sphere of radius r, where the flow structure is such that tracers show, e.g., strong variation in latitude and weak variation in longitude 共as is the case for atmospheric chemical species兲, an ‘‘equivalent latitude’’ coordinate ␾ e may be defined by A⫽2 ␲ r 2 (1⫺sin ␾e) 关see Fig. 1共b兲兴, and C may be regarded as a function of ␾ e and t. Similarly, in a three-dimensional volume of constant crosssectional area a, where tracers show strong variation in one direction, e.g., height, but weak variation in orthogonal directions, an ‘‘equivalent height’’ coordinate z e may be defined by V⫽az e 关see Fig. 1共a兲兴. In this last case, when considering density-stratified flows, the coordinate z e , based on density, may be used15 to express the minimum potential energy attainable through adiabatic rearrangement of a fluid as E b⫽g 兰 ␳ z edV and hence, the maximum amount of energy released under adiabatic rearrangement 共i.e., the available potential energy兲 as E b⫽g 兰 ␳ (z⫺z e)dV. B. Derivation of effective diffusion equation

In an incompressible flow (ⵜ•u⫽0), the time evolution of an advected scalar or tracer with concentration c(x,t) 共e.g., long-lived chemical species or Boussinesq density兲 is given by the advection-diffusion equation

⳵c ⫹u•ⵜc⫽ⵜ• 共 ␬ ⵜc 兲 , ⳵t

共3兲

where the scalar diffusivity is ␬ (x,t). This scalar diffusivity may be the molecular diffusivity or, in the case of numerical simulation, it may be an artificial diffusivity, included, for example, to ensure numerical stability. By making a transformation to isotracer coordinates, using the evolving tracer itself as the ‘‘reference’’ tracer 关i.e., c(x,t)⫽ ␹ (x,t)], one can reduce equation 共3兲 to a diffusion-only equation for the time evolution of the tracer as a function of area6 or volume7 共see Appendix for details兲. The transformation to tracer coordinates removes the advective term from the evolution equation 共3兲 for the tracer concentration and resulting equation simply describes the diffusion of tracer relative to the tracer contours themselves. For two-dimensional domains

冉

冊

⳵ C 共 A,t 兲 ⳵ ⳵ C 共 A,t 兲 ⫽ K eff共 A,t 兲 , ⳵t ⳵A ⳵A

共4兲

where K eff共 A,t 兲 ⫽

具 ␬ 兩 ⵜc 兩 2 典 A 共 ⳵ C/ ⳵ A 兲 2

,

共5兲

and a similar equation holds in three dimensions. Note that K eff is an effective diffusivity, although it does not have the usual dimensions of diffusion since A does not have the dimension of length.

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Phys. Fluids, Vol. 15, No. 11, November 2003

Diagnosing transport and mixing

C. Equivalent length or area

L eq⬃L⬃

For constant ␬, Eq. 共5兲 can be written as K eff共 A,t 兲 ⫽ ␬

具 兩 ⵜc 兩 2 典 A 共 ⳵ C/ ⳵ A 兲

2 2 ⫽ ␬ L eq共 A,t 兲 ,

共6兲

thereby defining the quantity L eq , the ‘‘equivalent length.’’ The equivalent length is related to the actual length of the tracer contour corresponding to concentration value C, i.e., L(C,t)⫽ 兰 ds, by 2 L eq ⫽

冕

␥ (C,t)

兩 ⵜc 兩 ds

冕

冏 冏

ds 1 ⫽L 2 兩 ⵜc 兩 ⭓L 2 , ⵜc ␥ (C,t) 兩 ⵜc 兩

共7兲

where (•)⫽ 兰 (•)ds/ 兰 ds denotes the line average around a tracer contour 共the inequality follows from Cauchy– Schwartz兲. Similarly, A eq , the ‘‘equivalent area,’’ may be defined for three-dimensional domains, and it can be shown 2 ⭓A 2 . Broadly speaking, the equivalent length or that A eq area is, therefore, greatest where the tracer contours or surfaces are largest relative to their minimum possible size, i.e., where the geometric structure of tracer is most complex. The ‘‘mixing ability’’ of a flow is essentially the ability to produce complex tracer geometry. This motivates the heuristic idea, exploited in previous studies, that the equivalent length or area may be used as a diagnostic of the flow, being relatively large in mixing regions and relatively small in barrier regions. It is this heuristic idea that we aim to make more quantitative in this paper. 关Note that because equivalent length or area purely characterize the tracer geometry, they are not changed by redefining the tracer concentration, i.e., ˜ ⫽ f (c), they are invariant under the transformation c哫c 2 2 2 ˜ since from Eq. 共7兲, L eq哫L eq⫽L eq , and similarly for K eff .] D. Scaling of equivalent length or area and effective diffusivity with diffusivity

In this section we present some further heuristic arguments, this time for the scaling with diffusivity of equivalent length and related quantities. Again, we aim to make these arguments more quantitative in later sections. In a two-dimensional steady flow it is known that for small enough diffusivity, after an initial period of adjustment, tracer contours align with streamlines 共e.g., Ref. 29兲. We speculate that, a similar adjustment occurs in a timedependent flow if it contains perfect barriers to advective transport, corresponding to so-called ‘‘invariant tori’’ 共see Sec. III C兲. We anticipate that in barrier regions the tracer contours will align with the invariant tori, and hence the length L of the tracer contours will be determined entirely by the geometry of those tori. If in addition ⵜc is bounded along such tori for all ␬, then from Eq. 共7兲 L eq will be bounded by some multiple of L, meaning L eq will also be determined entirely by the geometry of the tori and hence will be independent of diffusivity ␬. On the other hand, in a mixing region characterized by a stretching rate S with size A m⫽A m(S), the mixing region will be filled with tracer filaments of thickness L fS ⬃( ␬ /S) 1/2. Since L eq⭓L, for small ␬, one might expect

Am L fS

⬃A m

冉冊 S ␬

3345

1/2

.

共8兲

This scaling of equivalent length with stretching rate is consistent with the expectation that L eq will be large in regions of strong mixing, and small in regions of weak mixing. 关Note that it has been assumed that the stretching rate S is fixed over all scales, as is the case in the Batchelor regime of turbulence and in time-dependent flows where the velocity field is specified as a simple function of space and time, e.g., in the time-periodic flows that have been the subject of many chaotic advection studies. In flows that are not characterized by a stretching rate fixed over all scales, e.g., threedimensional 共3D兲 turbulent flows or more generally the nonsmooth flows discussed by Falkovich et al.,16 then the relevant S will depend itself on L and the relation is more complex.17,18兴 In the region of a perfect barrier, L eq will, according to the above arguments, be independent of ␬ and hence K eff will scale with ␬. This is consistent with recent calculations that show that the decay rates of the eigenmodes of the advection-diffusion problem for flows with perfect barriers tend to zero as ␬ tends to zero19 共and also, of course, with the fact that it is the presence of diffusion alone that allows the transport of tracer across a perfect barrier兲. In a mixing region we suggest, based on the above arguments, there is a small-diffusivity limit in which the effective diffusivity is largely independent of the diffusivity since, for small ␬ S 2 K eff⫽ ␬ L eq ⬃ ␬ A 2m ⫽SA 2m. ␬

共9兲

For flows with weakly permeable ‘‘partial’’ barriers 共i.e., barriers that are weakly permeable to advected particles in the absence of diffusion兲 there may be a more complex relation between ␬ and K eff . However, since the partial barrier is permeable to particles, it is to be expected that the transport across it 共and hence K eff) will be independent of ␬, if ␬ is small enough. Results will be presented in later sections that generally confirm the above arguments and estimates and show that, if the diffusivity is small enough, the qualitative structure, and to a large extent the magnitude, of K eff varies little with ␬ in mixing regions or partial barriers. E. Method for calculating effective diffusivity

The implication of the above discussion is that it may be possible to diagnose the physically relevant transport and mixing properties of a flow by considering a tracer in that flow and calculating the associated effective diffusivity. Presented with a flow velocity field, the technique is to first solve the advection-diffusion equation for a ‘‘test tracer’’ in the flow, with a suitably small choice of the diffusion parameter ␬. The tracer is initialized, taking into account the coherent structure in the flow, in such a way as to ensure that as the tracer evolves there is sufficient contrast in tracer values between different transport regions to allow a tracer-based coordinate system to be useful. The tracer field is extracted from the advection-diffusion simulation at regular time-

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Phys. Fluids, Vol. 15, No. 11, November 2003

intervals, and the effective diffusivity is calculated from it. At each time the gradient 兩 ⵜc 兩 2 of the tracer is calculated at all grid points; this is integrated over the area bounded by the desired tracer contour, 兰 c⭐C(A,t) 兩 ⵜc 兩 2 dA; and then this quantity is differentiated with respect to area, by taking finite differences, to obtain 具 兩 ⵜc 兩 2 典 A . This average is divided by the square of the gradient with respect to area of the tracer 2 ( ⳵ C/ ⳵ A) 2 , to obtain L eq (A,t) as defined in Eq. 共6兲. Then the 2 (A,t). effective diffusivity is calculated as ␬ L eq

III. DIAGNOSING THE TRANSPORT AND MIXING PROPERTIES OF A CHAOTIC ADVECTION FLOW A. Chaotic advection

In this section we present the effective diffusivity calculated for a time-dependent two-dimensional incompressible flow on a unit sphere (r⫽1) with streamfunction ␺ (␭, ␾ ,t), where ␭ is longitude and ␾ is latitude. We choose to study a flow that gives rise to either a perfect 共impermeable兲 or a partial 共permeable兲 barrier to advective transport dividing two mixing regions, with the permeability of the barrier being controlled by the parameters of the flow. The equations of motion of a fluid particle in such of flow form a Hamiltonian system, with Hamiltonian ␺. As is well known, even if the fluid velocity (u(␭, ␾ ,t), v (␭, ␾ ,t)) is a simple function of time, fluid particle trajectories (␭(t), ␾ (t)) can exhibit chaotic behavior, with neighboring trajectories separating exponentially fast in time, a phenomenon known as ‘‘chaotic advection.’’ 20 Generically, sets of nonchaotic 共i.e., integrable兲 trajectories, which separate linearly in time, divide the phase space into distinct chaotic regions. In the advection problem, the chaotic regions represent regions of strong fluid mixing, while the nonchaotic dividers represent perfect barriers to fluid particle transport. Chaotic advection studies have frequently investigated unidirectional shear flows subjected to time-periodic disturbances in the context of geophysics,21–24 plasma physics1,25 or both.26,27 Such flows are often obtained from linearizing the governing equations 共quasi-geostrophic or Hasegawa– Mima兲 and modeling the streamfunction or electrostatic potential by superposition of a small set of eigenmodes. Typically the unidirectional flow plus a single eigenmode defines a steady flow 共steady in some frame兲 and the addition of further eigenmodes gives nontrivial time dependence. A few studies26,27 have considered steady flows that violate the socalled ‘‘twist condition,’’ i.e., there is an interior streamline on which the frequency is maximum or minimum. This is generally due to the presence of a jet in the background unidirectional flow 共though it may also occur when the unidirectional flow is uniform兲. Although transport characteristics of flows satisfying the twist condition have been studied extensively, the transport characteristics of nontwist flows that violate the condition have been studied less and yet are interesting and can be rather subtle. Perfect and partial barriers to transport in these flow are often associated with the location of the jet, and the permeability of the partial barriers may be varied by altering the parameters that control the strength and position of the jet or the magnitude of the dis-

E. Shuckburgh and P. Haynes

turbances. 共Note that the transition between a perfect and a partial barrier is often rather complicated over a range of parameter space for nontwist flows.28兲 Following previous studies,21 we choose to investigate a time-periodic flow given by

␺ 共 ␭, ␮ ,t 兲 ⫽b ␮ ⫺a ␮ 3 ⫹ 共 1⫺ ␮ 2 兲 3/2 ⫻ 共 ⑀ 1 cos ␭⫹ ⑀ 2 cos共 ␭⫹ ␻ t 兲兲 ,

共10兲

with ␮ ⫽sin ␾, and a, b, ⑀ 1 , ⑀ 2 , and ␻ constant. The flow is steady when ⑀ 2 ⫽0. The velocity fields are defined in terms of ␺ by: u⫽U cos ␾,v⫽V cos ␾, where U⫽⫺ ⳵ ␺ / ⳵ ␮ ,V ⫽( ⳵ ␺ / ⳵ ␭)(1⫺ ␮ 2 ) ⫺1 . This flow violates the twist condition. 共We will be interested in the transport roughly across rather than along streamlines, and so the anomalous diffusion noted in some chaotic advection flows26,27 will not be relevant.兲 For ⑀ 2 ⫽0, the flow is integrable and particle trajectories are identical to streamlines. Figure 2共a兲 shows the variation in the streamfunction as the parameter b is altered. The el⫻ liptic (␭ •n , ␾ •n ) and hyperbolic (␭ ⫻ n , ␾ n ) critical points, i.e., instantaneous stagnation points, of the flow are indicated. Trajectories passing along the stable and unstable directions through the hyperbolic points separate the flow into different regions, and as the parameters of the flow are varied, these separatrices can undergo a topological change that may result in previously separated regions becoming connected. When b is decreased from b⫽0.35, one moves from a regime where there are two recirculation regions enclosed by homoclinic separatrices and a central jet which passes between them, through a reconnection (b⫽0.2235), to a new regime (b⫽0.05) with a distinct topology in which the central jet meanders strongly. 共As b is decreased still further, b⫽⫺0.1553, a tangent bifurcation occurs where the elliptic fixed points collapse into the hyperbolic fixed points and the recirculation regions are destroyed.兲 For ⑀ 2 ⫽0, the flow is not integrable and chaotic trajectories will appear in the vicinity of the separatrices of the instantaneous flow. As ⑀ 2 increases, the chaotic regions will enlarge and the non-chaotic regions will narrow, and hence in the advection problem the mixing regions will enlarge and the barrier regions narrow. The ability to alter the mixing/ barrier properties by varying a single parameter ⑀ 2 makes this family of flows particularly attractive for use as a testbench in a systematic study of the usefulness of effective diffusivity as a transport and mixing diagnostic. B. Effective diffusivity

1. Calculation of effective diffusivity

As has already been noted, when using the effective diffusivity formulation it is convenient to to define a new coordinate based on the area A within a tracer contour, but with dimensions of length. For flow on the sphere this coordinate is the equivalent latitude ␾ e defined by A⫽2 ␲ (1⫺sin ␾e ). Equations 共4兲 and 共5兲 can then be re-written with C as function of ␾ e and t, rather than A and t as

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Phys. Fluids, Vol. 15, No. 11, November 2003

Diagnosing transport and mixing

FIG. 2. 共a兲 Streamfunction for steady flow with parameters (a, ⑀ 1 , ⑀ 2 , ␻ ) ⫽(0.25,0.25,0,1) and various values of b 共hyperbolic critical points ⫻, elliptic critical points 䊉兲; and 共b兲 effective diffusivity for the same parameters (a, ⑀ 1 , ␻ ) and ⑀ 2 ⫽0.0125, with ␬ eff (⫻10⫺3 ) in grayscale. Axes are longitude ␭⫽ 关 0,2␲ 兴 (x axis兲 and latitude ␾ ⫽ 关 ⫺ ␲ /2 , ␲ /2兴 (y axis兲; note that spherical symmetry allows for the identification of ␭⫽0 with ␭⫽2 ␲ .

冉

冊

⳵ C 共 ␾ e ,t 兲 ⳵ ⳵ C 共 ␾ e ,t 兲 1 ⫽ 2 ␬ 共 ␾ ,t 兲 cos ␾ e , ⳵t r cos ␾ e ⳵ ␾ e eff e ⳵␾e 共11兲 where the effective diffusivity ␬ eff is defined by

␬ eff共 ␾ e ,t 兲 ⫽ ␬ r 2

具 兩 ⵜc 兩 2 典 A 共 ⳵ C/ ⳵ ␾ e兲

2⫽

2 ␬ L eq 共 ␾ e ,t 兲 . 2 ␲ r cos ␾ e兲 2 共

共12兲

Note that ␬ eff does have the usual dimension of diffusion 共whereas K eff did not兲. To obtain a tracer field from which the effective diffusivity may be calculated, a sample tracer is initialized as the steady part of the streamfunction. Initializing the tracer in this way minimizes the adjustment time during which the tracer aligns with the flow. In this case, the equivalent latitude does not simply correspond to latitude, instead the symmetry of the streamfunction means that ␾ e⫽0 corresponds to the position of a central jet, and ␾ e⫽⫾ ␲ correspond to the

3347

core of the recirculation regions. The advection-diffusion equation associated with the full unsteady flow is integrated using a spectral model to calculate the tracer field at later times. We run the integration with a range of small values of ␬, chosen to be large enough to inhibit the formation of features in the tracer at the grid-scale. A number of careful tests were performed to assess the sensitivity of the tracer calculation to numerical resolution. The tracer advection is not strongly sensitive to the resolution of the spectral model, except in that at higher model resolution a smaller value of ␬ is sufficient to prevent structure at the grid scale. The tracer advection is also insensitive to the timestep used in the model, within a significant range. In this section we present results from a simulation where we used a spectral truncation of T255 共i.e., spherical harmonics with a total wave number less than 255, corresponding to a grid resolution of about 0.01 radians兲 as a resolution that allows for small ␬ ( ␬ ⫽5 ⫻10⫺6 ) but is not too computationally expensive. A time step of ␶ /1200 was chosen, where ␶ ⫽2 ␲ / ␻ is the period of the flow. Once a period, ␬ eff(␾e ,t) was calculated directly from the tracer field. The effective diffusivity for flows with perturbation ⑀ 2 ⫽0.0125 and various values of b is shown in Fig. 2共b兲 for t⫽30␶ . The value of ␬ eff associated with each tracer contour c(x,t)⫽C is plotted, making use of the relation between C and ␾ e . It is important to note that ␬ eff is constant along a tracer contour, and so the two-dimensional ␬ eff plots shows only one-dimensional information; the two-dimensional structure comes solely from the tracer distribution c(x,t). In practice, the mixing properties might vary significantly along a tracer contour, however the effective diffusivity, which is an average quantity, will not represent this. In Fig. 2共b兲 regions of strong mixing are represented by large values of ␬ eff 共white兲, and barrier regions are represented by small values of ␬ eff 共black兲. Comparing Figs. 2共a兲 and 2共b兲, it can be seen that, for each case where there is a jet in the flow, the region of the jet is indicated to be a barrier region by effective diffusivity, and where there are regions of strong rotational motion, they are characterized by weak mixing. Around the separatrices of the steady streamfunction there are regions of enhanced mixing. From now on we will study exclusively the flow with parameters (a, ⑀ 1 , ␻ ) as above and b⫽0.05, since in this case the mixing properties are expected to depend in a relatively simple way on ⑀ 2 , with the mixing regions around the separatrices growing in strength and size as ⑀ 2 is increased. We consider 22 values of ⑀ 2 in the range 0⭐ ⑀ 2 ⭐0.2. For these parameters the topology of the streamfunction is always such that the central jet meanders through the domain. The effective diffusivity at t⫽30␶ for various values of ⑀ 2 is shown in Fig. 3. As expected, for small ⑀ 2 , such as ⑀ 2 ⫽0.0125, two weak ‘‘mixing regions’’ ( ␬ eff⬇0.625⫻10⫺4 ) form in the vicinity of the separatrices. The mixing regions are bounded by ‘‘barrier regions’’ ( ␬ eff⬇0) at the central jet and the cores of the recirculation regions. As ⑀ 2 increases,

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E. Shuckburgh and P. Haynes

FIG. 3. Effective diffusivity for various ⑀ 2 , ␬ eff (⫻10⫺3 ) in grayscale.

the size and intensity of the mixing region increases ( ␬ eff ⭓0.5⫻10⫺3 at ⑀ 2 ⫽0.075) and the barriers narrow and weaken ( ␬ eff⬇0.625⫻10⫺4 for the central barrier at ⑀ 2 ⫽0.075). It is interesting to note that considerable inhomogeneity exists throughout the mixing region for all values of ⑀ 2 . The transport and mixing properties diagnosed here by ␬ eff will be compared in later sections with that obtained from Poincare´ sections and from Lyapunov exponents. Figure 4 shows the equivalent length 共solid兲 as a function of equivalent latitude for ⑀ 2 ⫽0.05 and ⑀ 2 ⫽0.15. The strength of the central barrier is indicated by the value of L eq at ␾ e⫽0, and it can be seen that this strength reduces considerably 关 L eq(0) increases兴 as ⑀ 2 increases. Figure 4 confirms that the equivalent length is always greater than or

equal to the actual length 共dash兲 of a tracer contour 关Eq. 共7兲兴, and that the two quantities are closely related. Also shown in Fig. 4 is the variation of L eq with ⑀ 2 . The value of L eq in the mixing region 共light gray兲 increases almost steadily as ⑀ 2 increases. The value of L eq at the central barrier 共dark gray兲 stays small until about ⑀ 2 ⫽0.075, after which it increases steadily, indicating substantial weakening of the barrier. If effective diffusivity is a well-defined diagnostic of the flow, it should depend principally on the details of the flow field, and not, for example, on the initial condition of the tracer field or the imposed numerical diffusivity. For a steady two-dimensional flow, with small values of the diffusivity, there is a straightforward relationship between the effective diffusivity and the cross-streamline diffusivity calculated by

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Diagnosing transport and mixing

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FIG. 4. L eq( ␾ e) 共solid兲 and L( ␾ e) 共dash兲 for 共a兲 ⑀ 2 ⫽0.05 and 共b兲 ⑀ 2 ⫽0.15; and 共c兲 L eq for all 22 values of ⑀ 2 for the mixing region (0.28⭐ ␾ e⭐0.7, light gray兲, and the barrier region (0.09⭐ ␾ e⭐0.09, dark gray兲, as indicated by vertical lines in the other plots. Note ␾ e ⫽⫾ ␲ /2 corresponds to the center of the recirculation regions and ␾ e ⫽0 to the central jet.

Rhines and Young.29,30 Since the Rhines and Young diffusivity is defined without reference to any tracer field, it follows that for a steady flow, the effective diffusivity can indeed be regarded as a well-defined diagnostic of a steady flow. We now present the results of various experiments designed to test whether ␬ eff is a well-defined diagnostic of a timedependent flow. 2. Dependence on initial tracer field

To test the dependence of effective diffusivity on the choice of initial tracer field, we repeated the calculation of ␬ eff with ⑀ 2 ⫽0.1 for various different initial conditions c(␭, ␾ ,0)⫽c 0 (␭, ␾ ) for the tracer, chosen to give high and low gradients in different regions. Figure 5 shows ␬ eff(␾e,30␶ ) for four cases, c 0 ⫽ ␺ , 10␺ , 23 sin(␺/␺max) and sin3(␲␺/2␺ max), where ␺ max is the maximum value of ␺ at t⫽0. For c 0 ⫽ ␺ , 10␺ , and 32 sin(␺/␺max) 共black curves兲 the effective diffusivity is identical at t⫽30␶ 共although there are significant differences in gradient of the tracer兲. However, for the other case (c 0 ⫽sin3 (␲␺/2␺ max), gray curve兲, anomalously large values of ␬ eff are found both poleward of ⫾ ␲ /4 and at the equator. The large ␬ eff values coincide with regions of almost zero gradient with respect to ␾ e in the initial tracer profile. This is unsurprising since in this case both the numerator and denominator of Eq. 共12兲 will be small and, therefore, the evaluated expression will be sensitive to small numerical errors in either, and furthermore the assignment of equivalent latitude breaks down in regions of zero tracer gradient. 共For long-time integrations it may, therefore, be sensible to include a weak source term in the calculation of the evolution of the tracer to prevent regions of almost zero tracer gradient developing in the mixing regions.31兲 We con-

FIG. 5. 共a兲 Four different tracer initial conditions, c 0 ⫽ ␺ 共solid 3 black兲, c 0 ⫽10␺ 共dash–dot black兲, c 0 ⫽ 2 sin(␺/␺max) 共dash black兲, 3 c 0 ⫽sin (␲␺/2␺ max) 共dot gray兲, and 共b兲 the corresponding ␬ eff at t⫽30␶ .

clude that the effective diffusivity can be considered, for practical purposes, independent of the initial conditions of the tracer field, so long as those initial conditions do not contain regions of almost zero gradient with respect to ␾ e . 3. Evolution in time

In a time-periodic chaotic advection flow a tracer will develop patterns whose amplitude decays slowly with time but whose geometric structure persists and indeed repeats from one period to the next.32–34 In Fig. 6 it can be seen that while the tracer geometry changes substantially within a period (t⫽29.25␶ ,29.5 ␶ ,29.75␶ ), it changes little from one period to the next (t⫽29␶ ,30␶ ). One expects, therefore, that after an initial adjustment phase lasting a few eddy turnover times where the tracer aligns with the flow, the effective diffusivity, which is determined by the tracer geometry, will also vary little from one period to the next. This is demonstrated for ⑀ 2 ⫽0.05,0.15 in Fig. 7. Considering all 22 values of ⑀ 2 , the average difference in ␬ eff for the mixing region (0.28⬍ 兩 ␾ e兩 ⬍0.7, light gray兲 and the barrier region ( 兩 ␾ e兩 ⬍0.09, dark gray兲 between t⫽25␶ and t⫽30␶ is almost always less than 10%. Figure 6 shows further that the effective diffusivity changes little within a period, indicating that although the geometric alignment of the tracer changes within a period, the ‘‘complexity’’ of it, as measured by L eq , does not. We conclude that ␬ eff reaches a quasi-steady state after a few eddy turnover times. 4. Dependence on diffusivity

Above, we suggested that, for small enough ␬, ␬ eff may not depend on ␬ within mixing regions and partial barriers. We now investigate whether our calculation of ␬ eff is sensitive to our choice of ␬ ⫽5⫻10⫺6 . Figure 8 shows ␬ eff(␾e) for various values of ⑀ 2 with ␬ ranging over an order of magnitude from 4⫻10⫺6 共solid兲 to 4⫻10⫺5 共dot兲. It can be seen that the latitudinal structure of ␬ eff does not alter greatly as ␬ is varied over this range. Figure 9 shows values of ␬ eff averaged over the mixing regions 共light gray兲 and over the central barrier region 共dark gray兲 for ␬ varying over three orders of magnitude. In the mixing region, in particular for ⑀ 2 ⫽0.1 and 0.15, it can be seen that ␬ eff varies little 共less than one order of magnitude兲 with ␬. This is consistent with the scaling in Eq. 共9兲. Where there is a strong barrier at ⑀ 2 ⫽0.05, ␬ eff clearly depends strongly on ␬ for larger values of ␬, although it appears to be tending to a finite, non-zero limit as ␬ tends to 0. Again this is consistent with the heuristic

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Phys. Fluids, Vol. 15, No. 11, November 2003

E. Shuckburgh and P. Haynes

FIG. 6. Effective diffusivity for ⑀ 2 ⫽0.1 at various time intervals within a period 共a兲–共e兲, and 共f兲 as a function of ␾ e for each time. Scales are ⫻10⫺3 .

arguments given above. We conclude, therefore, that for mixing regions and partial barriers, ␬ eff does seem to tend to a finite, nonzero limit as ␬ tends toward zero. Furthermore, we believe that calculations with our chosen value of ␬ ⫽5 ⫻10⫺6 , will provide a reasonable estimate of this limit. 共Note that Fig. 9 also demonstrates that the value of ␬ eff is not sensitive to the resolution of the tracer calculation.兲

5. Effective diffusivity is a well-defined diagnosticof the flow

In this section we have demonstrated that the effective diffusivity is a reasonably well-defined diagnostic of the flow: 共i兲 Being reasonably independent of the tracer initial conditions, providing the tracer gradients are not too weak, 共ii兲 reaching a quasi-steady state after a few eddy turnover times, 共iii兲 being reasonably independent of the diffusivity ␬, except within near-perfect barriers.

FIG. 7. Effective diffusivity (⫻10⫺3 ) for 共a兲 ⑀ 2 ⫽0.05 and 共b兲 ⑀ 2 ⫽0.15 at t⫽20␶ 共dot兲, 25␶ 共dash兲, and 30␶ 共solid兲.

C. Poincare´ sections

We now investigate the advective transport properties of the chosen family of flows, as revealed by considering the associated Poincare´ sections 共see Drazin,35 Chap. 8兲. We will compare the transport and mixing structure so determined with that indicated by the effective diffusivity in Fig. 3. Considering the Poincare´ section for different points in a flow, quasi-periodic orbits of particles that map out solid lines known as ‘‘invariant tori’’ denote perfect barriers to advective transport that separate the flow into distinct regions, whereas orbits of particles that are area-filling denote chaotic regions of advective mixing. Moreover, variations to the transport properties within a family of flows can be explained in terms of topological changes36 –38 to the associated Poincare´ sections, which are often rather generic within broad categories of flows, such as the category of ‘‘nontwist’’ flows to which our chosen family belongs.26,27,39– 41 Figure 10 shows the Poincare´ sections for the streamfunction 共10兲 with different values of the parameter ⑀ 2 . The Poincare´ sections have two symmetries: 共i兲 Invariance to translation by ␲ in longitude, followed by reflection about the ␾ ⫽0, and 共ii兲 invariance to reflection about longitude ␭⫽ ␲ . The first symmetry is shared by particle trajectories, and by a tracer field 共and hence ␬ eff ; see Fig. 3兲. The second symmetry, due to invariance under time reversal, only applies to the Poincare´ sections. Each panel in Fig. 10 is a composite of the Poincare´ sections for three different initial conditions: The light-gray particles were initialized at the mid-point between the two elliptic points 共i.e., on the central barrier兲, and the mid- and dark-gray particles were initialized at locations along the line connecting the elliptic points of the steady streamfunction. The later positions of these particles are plotted for 10 000 periods.

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Phys. Fluids, Vol. 15, No. 11, November 2003

Diagnosing transport and mixing

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FIG. 8. Dependence of effective diffusivity with diffusivity. Vertical axis is ⫻10⫺3 . Curves are for ␬ ⫽4 ⫻10⫺6 共solid兲, ␬ ⫽1.2⫻10⫺6 共dash兲, and 4⫻10⫺5 共dot兲.

As ⑀ 2 is increased from zero 共see ⑀ 2 ⫽0.0125) chaotic regions 共mid-gray兲 form in the vicinity of the separatrices of the steady flow, as expected, and as indicated by ␬ eff in Fig. 3. The chaotic region is bounded by nonchaotic invariant tori: The dark-gray torus lies in the recirculation region, and the light-gray torus divides the two distinct chaotic regions 共the second chaotic region is not shown but is inferred by symmetry兲 and lies within the barrier region of the jet. For ⑀ 2 ⫽0.025 the two distinct chaotic regions are enlarged and the central torus 共light-gray兲 is distorted, indicating a nearby period-7 orbit. The central barrier, and its distortion, can be observed in ␬ eff 共Fig. 3兲. As ⑀ 2 is increased to ⑀ 2 ⫽0.05, there is a topological change with the break-up of the central torus and the merging of the two chaotic regions to form one large region defined by the area visited by light- and midgray particles. 共Note that as a consequence of the finite time for which the Poincare´ sections have been generated, two distinct sub-regions are still discernible, one which happens to have more–less light- to mid-gray points, however, were the Poincare´ sections to be plotted for longer and longer times, the difference between the two sub-regions would be less and less obvious.兲 This implies that trajectories cross the boundary between the two sub-regions relatively infrequently and, therefore, that this boundary may be considered as a partial barrier to advective transport. The minimum in ␬ eff picks out the shape of this partial barrier with remarkable accuracy. Furthermore, within the chaotic region, islands of nonchaotic behavior can be observed, and these are clearly represented by regions of small ␬ eff . For still larger perturbation of the flow, with ⑀ 2 ⫽0.075, there are still two subregions within the chaotic region, but the difference between the subregions is impossible to distinguish at this time 共10 000 periods兲. This implies that the time scale for crossing the partial barrier has reduced from that for ⑀ 2 ⫽0.05, and is now, presumably, considerably less than 10 000 periods; this weakening of the partial barrier is apparent in ␬ eff . This com-

parison of Poincare´ sections and ␬ eff for different flow regimes demonstrates that ␬ eff can accurately identify mixing regions and partial barriers in a flow. D. Relation of effective diffusivity to particle transport

In the previous section, we used a tool for analyzing the chaotic behavior of advective transport to conclude that the effective diffusivity is a useful diagnostic of the qualitative structure of transport and mixing. In this section we assess the usefulness of effective diffusivity for quantifying transport and mixing. We again chose to take the purely advective case for comparison, and we focus in particular on transport across the central barrier. We compare particle transport with solution of the diffusion equation 共11兲, interpreting the concentration C as the density of particles measured with respect to equivalent latitude coordinates. Note that we have no a priori reason to believe that the advective transport of particles should correspond in any quantitative sense to the transport of tracer subjected to both advection and small diffusion. We adopt this approach simply as a modeling hypothesis—we have no theoretical basis for suggesting that Eq. 共11兲 should precisely capture the transport of advected particles, nevertheless one might expect some similarity for small diffusion. To investigate the purely advective case, approximately 8000 particles were placed on a regular grid 共with spacing ␦ ␭⫽ ␲ /100 and ␦ ␾ ⫽ ␲ /200) within the region of equivalent latitude ␲ /18 ⭐ ␾ e⭐ 4 ␲ /18 chosen to lie within one of the chaotic mixing regions for all values of ⑀ 2 to be investigated, a distance away from the central barrier and outside the weakly mixed recirculation regions. Figure 11 shows the region in which the particles were initialized 共white兲, and the position of those particles that have crossed the barrier 共as

FIG. 9. Dependence of effective diffusivity with diffusivity. Values of effective diffusivity in the mixing region (0.28⬍ 兩 ␾ e兩 ⬍0.7, light gray兲 and in the barrier region ( 兩 ␾ e兩 ⬍0.09, dark gray兲. Calculations at resolution: T85 共square兲, T170 共asterisk兲, T255 共diamond兲, and T341 共cross兲.

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Phys. Fluids, Vol. 15, No. 11, November 2003

E. Shuckburgh and P. Haynes

FIG. 10. Poincare´ sections for increasing values of ⑀ 2 . Each plot is a composite of the Poincare´ sections for three different initial conditions, spaced along the line connecting the elliptic points.

defined by the minimum in effective diffusivity at ␾ e⫽0) from ␾ e⬎0 to ␾ e⬍0 after 10␶ 共black dots兲. As expected, for ⑀ 2 ⫽0.05, when the barrier is strong, no particles have crossed the barrier after 10 periods, whereas for ⑀ 2 ⫽0.15,

when the barrier is weak, about 1500 particles 共almost 20%兲 have crossed the barrier. The particle distribution relative to tracer contours for each ⑀ 2 can be expressed by a probability density function

FIG. 11. Position of particles 共black dots兲 that have crossed the central barrier 共black contour兲 after t⫽10␶ . Initialization region for particles indicated in white, equivalent latitude contours in grayscale.

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Phys. Fluids, Vol. 15, No. 11, November 2003

Diagnosing transport and mixing

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FIG. 12. Particle density 共dot兲 calculated from approximately 8000 particles initialized in the region ␲ /18 ⭐ ␾ e⭐ 4 ␲ /18 on a regular grid 共with spacing ␦ ␭ ⫽ ␲ /100 and ␦ ␾ ⫽ ␲ /200), for t⫽0 (black), t⫽10␶ 共dark gray兲, t⫽20␶ 共mid-gray兲 and t⫽50␶ 共light gray兲, scale on left axis. Overlaid is the predicted pseudo particle density 共solid兲 derived from integrating a diffusion equation with diffusivity ␬ eff , colors and scale as for pdf. Dashed line shows ␬ eff , scale on right axis (⫻10⫺3 ).

共calculated using a fixed-width kernel density estimation兲. This gives the actual particle density in equivalent latitude coordinates. We also predict a pseudo particle density by solving the diffusion equation 共11兲, using a Crank– Nicholson scheme, for a tracer with the initial distribution given by the actual particle density at t⫽0. The value of ␬ eff(␾e) is taken to be that calculated for the various ⑀ 2 in earlier Sec. III B. Figure 12 shows the actual particle density 共dot兲 and the predicted pseudo particle density 共solid兲 based on the effective diffusivity 共dash兲, at various times t⫽0, 10␶, 20␶, 50␶. It can be seen that there is remarkably good quantitative agreement at each time for all values of ⑀ 2 . So, the distribution of advective particles seems to be well modeled in a quantitative sense by the effective diffusivity derived from an advected and diffused tracer. We emphasize again

that we have no precise theoretical understanding of why this should be the case. We now take this idea still further. For the flow under consideration, the transport properties are dominated by the strength of the central partial barrier. It seems plausible, therefore, that the particle distribution might be approximated by a model with a partial barrier allowing leakage at constant rate ␯. Under this model if there are initially N particles on one side of the barrier, the number of particles x to have crossed to the other side at a later time t will be given using this model by

x⫽

N 共 1⫺e ⫺2 ␯ t 兲 . 2

共13兲

FIG. 13. 共a兲 For different ⑀ 2 , percentage of particles to have left initial chaotic region across the central barrier, with overlaid the best-fit to Eq. 共13兲; 共b兲 leakage rate for all 22 values of ⑀ 2 as estimated from particle model 共13兲 ␯, and as estimated from effective diffusivity model ␯ eff .

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Phys. Fluids, Vol. 15, No. 11, November 2003

Figure 13共a兲 shows the percentage of particles that have crossed the partial barrier at each time 共gray兲. The value of leakage rate ␯ has been chosen for each ⑀ 2 to give the best-fit of Eq. 共13兲 to these data 共black兲. Interpretation of the leakage rate depends on the assumption that advection within the chaotic region is rapid compared to transport across the partial barrier, and therefore, that it does not matter where particles are initialized within the mixing region, because the time as evaluated above will be dominated by the time to cross the barrier. We tested this assumption by choosing different sub-regions of the mixing region in which to initialize the particles and found differences in the calculated leakage rate of not more than 5%. 共Note that if we believe that the effective diffusivity can give us some quantitative indication of the transport of advective particles, then the fact that the effective diffusivity is large throughout the initialization region, and small throughout the partial barrier region tells us in itself that advection within the chaotic region is rapid compared to transport across the partial barrier.兲 A similar experiment to investigate particle transport across the shearless curve of a nontwist map 共which corresponds to our partial barrier兲 has been conducted by Corso and Rizzato.42 They related the transmission of particles to manifold reconnection-like processes, showing that as the relevant parameter is increased to enter a more chaotic regime, rearrangements of the unstable manifolds create new channels for transport.43 We suggest that if the effective diffusivity is able to provide a quantitative indication of the transport of advected particles, then perhaps it can be used to predict the leakage rate calculated above. To test this we assume ␬ eff has a small * in the barrier region, which is of width l b . constant value ␬ eff If a tracer has constant concentration values C 1 and C 2 in the two mixing regions, assumed to be of size l m , on either side of the barrier, then the flux across the barrier may be esti* /lb . Hence, the time scale on which mated as (C 1 ⫺C 2 ) ␬ eff C 1 ⫺C 2 approaches zero, the relaxation time scale, may be * from Eq. 共11兲. We estimate the widths estimated as l ml b/2␬ eff of the mixing region and barrier by considering the division between mixing region and barrier to occur where the magnitude of the gradient of ␬ eff is greatest. We then calculate * /lml b) predicted by the above arthe leakage rate ( ␯ eff⫽2␬eff gument. This is plotted in Fig. 13共b兲 against the leakage rate derived above 共␯兲. There is good agreement between these two estimates, for all 22 values of ⑀ 2 . We, therefore, conclude that first a highly simplified model of particle transport based on effective diffusivity has considerable skill, as indicated by Fig. 12, and second that effective diffusivity is a remarkably useful quantifier of advected particle transport, as indicated by Fig. 13. E. Relation of effective diffusivity to Lyapunov exponents

A commonly used diagnostic of mixing that, like effective diffusivity, may be applied to either time-periodic or aperiodic flows is the finite-time Lyapunov exponent44,45 共FTLE兲. In a chaotic region there is a distribution of Lyapunov exponents S⫽S(␭, ␾ ,T) at any finite time T. It is

E. Shuckburgh and P. Haynes

not clear how the distribution of FTLEs relates to mixing of a tracer field, nevertheless, this diagnostic has be applied to examine transport and mixing in both atmospheric45– 48 and oceanic49–52 flows. There are known to be various shortcomings of the FTLE approach53,54 and recently it has been suggested12,55 that there are advantages to using finite-size Lyapunov exponents56,57 共FSLE兲 instead of FTLE for identifying barriers.12,53 However, we leave comparison of FSLE and effective diffusivity to another paper. We calculate FTLE using the so-called pull-back method44 to remove nonlinear effects due to amplification. For each point we chose two small initial perturbation of length 10⫺6 in latitude and longitude and rescale these perturbations by a factor 0.1 if at any time step they grow greater than 10⫺5 in length. 共The calculation is not particularly sensitive to the choice of these parameters.兲 The FTLE are calculated on points of a T341 grid 共1024 points in longitude and 512 points in latitude兲. Direct computation of the FTLE is only made for half the points, with symmetry being invoked to infer the FTLE for the remaining points. Here we present the results of calculations of FTLEs made at time T ⫽10␶ . We consider the backward-time Lyapunov exponents since it is the past history of stretching that will influence the tracer structure. Looking to Fig. 14 it can be seen that there is considerable correspondence between the structures in the backwardtime FTLE 共a兲 and in the effective diffusivity 共b兲, for all values of ⑀ 2 . Both quantities indicate the presence of a weak-mixing regions of similar shape, located in the vicinity of the central jet and in the recirculation regions, and chaotic mixing regions which increase in size as ⑀ 2 increases. Figure 15 shows 共dash兲 K eff and 共solid兲 the FTLE at T ⫽10␶ averaged along each equivalent latitude 具 S 典 ␾ e multiplied by the area of the mixing region squared A 2m 共defined as size of region with FTLE greater than a threshold value of 0.05兲, with a view to examining the proposed scaling 共9兲. It can be seen that the lowest values of both K eff and of 具 S 典 ␾ eA 2m correspond to ␾ e⫽⫾ ␲ /2, and that there is a local minimum in both at ␾ e⫽0. However, there are some differences in the details of the distribution of K eff and 具 S 典 ␾ eA 2m within the mixing region 共these differences will be analyzed in a future paper兲. Furthermore, while K eff clearly indicates the presence of a weak central barrier at ⑀ 2 ⫽0.15, it is less obvious from the FTLE 共indeed at ⑀ 2 ⫽0.2, not shown, 具 S 典 does not indicate a central barrier at all兲. If a fluid element is initialized in a rather permeable barrier, one might expect that the time scale for that fluid element to leave the barrier region is short. If this departure time scale 关see Fig. 共11兲兴 is less than the time scale necessary to define the Lyapunov exponent 共here 10␶兲 then the FTLE will not simply represent the stretching in the barrier region. The FTLE may, therefore, not be a good tool to investigate the strength of permeable barriers. It is worth noting from Figs. 14 and 15 that the equivalent latitude coordinate is a useful way to organizing the mixing quantification so as to clearly identify the central barrier, which in latitude-longitude space is highly convoluted for large ⑀ 2 . m 2 /( ␬ L min ) Figure 16 shows the scaling of N eff⫽Keff

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Phys. Fluids, Vol. 15, No. 11, November 2003

Diagnosing transport and mixing

3355

FIG. 14. 共a兲 Finite-time Lyapunov exponents at T⫽10␶ and 共b兲 effective diffusivity 共lower panel兲 for various ⑀ 2 , scale is ⫻10⫺3 for ␬ eff .

m 2 2 2 2 ⫽(Leq ) /L min with N S ⫽A m S m /( ␬ L min ) for the mixing regions of flows with ⑀ 2 ⫽0.05, 0.075, 0.1, 0.15, 0.2, where L min⫽2␲ cos ␾e . Here we have defined the ‘‘mixing region’’ to be the region where the FTLE is greater than 0.05, and m m K eff , L eq , and S m are average values in this region. For N S m greater than about 100 共i.e., for small values of ␬兲, K eff is approximately independent of ␬, and scales almost linearly with A 2mS m , as suggested by the scaling 共9兲. As expected, for 2 2 /L min tends to a value of 1. large ␬, L eq

IV. DISCUSSION

The usefulness of the effective diffusivity as a potential diagnostic of transport and mixing has already been demonstrated in a number of practical situations arising from geophysics.8 –11,30,31,58 – 60 However, all these studies have re-

lied on heuristic arguments to assert that the effective diffusivity may be used to characterize transport and mixing. In this paper we have used a family simple chaotic advection flow to investigate in an objective, systematic manner the diagnostic capabilities of effective diffusivity, to demonstrate how it relates to other diagnostics of transport and mixing, and to discuss further its theoretical basis. We have conducted extensive experiments to demonstrate that the effective diffusivity is a well-defined flow diagnostic, i.e., determined by the flow, not the particular choice of reference tracer. We have demonstrated that the effective diffusivity is a particularly valuable measure of the strength of barriers, and moreover that it is able to identify many of the fine details of the transport and mixing structure. We have shown that the dispersion of particles across tracer contours is well

FIG. 15. A 2m具 S 典 ␾ e 共black solid, right axis兲, and K eff 共dash, left axis兲 for various values of ⑀ 2 .

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Phys. Fluids, Vol. 15, No. 11, November 2003

E. Shuckburgh and P. Haynes

sions. This research has been supported by a Marie Curie Fellowship of the European Community program ‘‘Improving Human Research Potential and Socio-economic Knowledge Base’’ under Contract No. HPMF-CT-2000-00636. Computing resources were funded by the U.K. Natural Environment Research Council through the U.K. Universities Global Atmospheric Modeling Program. E.F.S. would also like to acknowledge the support of Darwin College, Cambridge. APPENDIX: DERIVATION OF EFFECTIVE DIFFUSITY

FIG. 16. N eff plotted against N S , with K eff and S taken to be the average values in the mixing regions for ⑀ 2 ⫽0.05, 0.1 0.15, 0.2. Calculations at resolution: T85 共square兲, T170 共asterisk兲, T255 共diamond兲, and T341 共cross兲. The solid line has gradient 1.

modeled by diffusion, with the relevant diffusivity being the effective diffusivity, and hence that the effective diffusivity can quantify the leakage rate across a barrier. We note, however, that there is no precise theoretical understanding of why this seems to be the case, and highlight this as an open area of research. In some sense, the effective diffusivity can be considered a hybrid Eulerian–Lagrangian diagnostic, which takes into account the influence on a tracer of recent particle separation rates and assigns some measure of this influence to a particular location defined by tracer surfaces or contours. There is, therefore, some similarity between the effective diffusivity and the Lagrangian separation rates measured by FTLEs, but there are also some important differences. Most notably, it seems that in the presence of a narrow, partial barrier to transport the FTLEs fail to clearly identify the barrier where the effective diffusivity succeeds, and furthermore, the FTLEs do not take into account the size of the chaotic mixing region and hence do not properly characterize the increased transport that occurs as the disturbance to the flow increases. We believe that the effective diffusivity has major advantages as a quantitative diagnostic of transport and mixing of the aperiodic flows of importance in many geophysical, plasma physics and chemical engineering problems. We still do not have a clear theoretical understanding of the mathematical relationship between the effective diffusivity and particle transport. However, the results of this objective and systematic study support the use of effective diffusivity as a transport and mixing diagnostic, and thereby strengthen the conclusions of the above mentioned applications of effective diffusivity in real geophysical situations and support its use as a diagnostic in a wider context. ACKNOWLEDGMENTS

We are particularly grateful to Warwick Norton for providing assistance with the numerical computations and thank Bernard Legras and Noboru Nakamura for useful discus-

In this appendix we demonstrate how the tracer-based formulation may be used to derive an effective diffusivity. We focus on the case of a two-dimensional flow, however the results also hold in the three-dimensional case if one replaces ‘‘contour’’ and ‘‘area’’ with ‘‘surface’’ and ‘‘volume,’’ dA with dV, and ds with ds 1 ds 2 . First we obtain an equation for the rate of change in area within a tracer contour

⳵ A 共 C,t 兲 ⳵ ⫽ ⳵t ⳵t ⫽

冕

dA

c⭐C

冕

␥ (C,t)

uC •

ⵜc ds⫽⫺ 兩 ⵜc 兩

冕

⳵ c ds , ␥ (C,t) ⳵ t 兩 ⵜc 兩 共A1兲

C

where the velocity u is the velocity of the contour C that is defined, up to an arbitrary vector perpendicular to ⵜc, by ⳵ c/ ⳵ t⫹uC •ⵜc⫽0. Substituting for ⳵ c/ ⳵ t from Eq. 共3兲 we get

⳵ A 共 C,t 兲 ⫽ ⳵t

冕

␥ (C,t)

共 u•ⵜc 兲

冕

ds ⫺ 兩 ⵜc 兩

␥ (C,t)

ⵜ• 共 ␬ ⵜc 兲

ds . 兩 ⵜc 兩 共A2兲

The line integrals on the right-hand side of this equation can be written as a differential of an area integral using Eq. 共1兲. Then the divergence theorem can be applied 共noting that ⵜc/ 兩 ⵜc 兩 is the normal to ␥ (C,t)). In this way, the first term, the advective term, can be shown to be zero because the flow is incompressible. Only the second, diffusive term then remains

冕

⳵ A 共 C,t 兲 ⳵ ⫽⫺ ⳵t ⳵C

␥ (C,t)

␬ 兩 ⵜc 兩 ds.

共A3兲

Then we convert this expression for ⳵ A/ ⳵ t to one for ⳵ C/ ⳵ t using the chain rule for differentiation of C(A,t) with respect to t,

⳵ C 共 A,t 兲 ⳵ A 共 C,t 兲 ⳵ C 共 A,t 兲 ⫽⫺ ⳵t ⳵t ⳵A ⫽

冉

冊

⳵ ⳵ C 共 A,t 兲 K eff共 A,t 兲 , ⳵A ⳵A

共A4兲

where

⳵A K eff共 A,t 兲 ⫽ ⳵C

冕

␥ (C,t)

␬ 兩 ⵜc 兩 ds⫽

具 ␬ 兩 ⵜc 兩 2 典 A 共 ⳵ C/ ⳵ A 兲 2

,

共A5兲

following from the definition in Eq. 共2兲.

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Phys. Fluids, Vol. 15, No. 11, November 2003 1

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