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Stirring and mixing in two-dimensional divergent flow

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Lukovich, J. V. and Shepherd, T. G. (2005) Stirring and mixing in two-dimensional divergent flow. Journal of the Atmospheric Sciences, 62 (11). pp. 3933-3954. ISSN 1520-0469 doi: https://doi.org/10.1175/JAS3580.1 Available at http://centaur.reading.ac.uk/32103/

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Stirring and Mixing in Two-Dimensional Divergent Flow

JENNIFER VERLAINE LUKOVICH* AND THEODORE G. SHEPHERD

Department of Physics, University of Toronto, Toronto, Ontario, Canada

(Manuscript received 24 February 2005, in final form 25 April 2005)

ABSTRACT

While stirring and mixing properties in the stratosphere are reasonably well understood in the context ofbalanced (slow) dynamics, as is evidenced in numerous studies of chaotic advection, the strongly enhancedpresence of high-frequency gravity waves in the mesosphere gives rise to a significant unbalanced (fast)component to the flow. The present investigation analyses result from two idealized shallow-water nu-merical simulations representative of stratospheric and mesospheric dynamics on a quasi-horizontal isen-tropic surface. A generalization of the Hua–Klein Eulerian diagnostic to divergent flow reveals that velocitygradients are strongly influenced by the unbalanced component of the flow. The Lagrangian diagnostic ofpatchiness nevertheless demonstrates the persistence of coherent features in the zonal component of theflow, in contrast to the destruction of coherent features in the meridional component. Single-particlestatistics demonstrate t2 scaling for both the stratospheric and mesospheric regimes in the case of zonaldispersion, and distinctive scaling laws for the two regimes in the case of meridional dispersion. This is incontrast to two-particle statistics, which in the mesospheric (unbalanced) regime demonstrate a more rapidapproach to Richardson’s t3 law in the case of zonal dispersion and is evidence of enhanced meridionaldispersion.

1. Introduction

Fluid motion in the extratropical stratosphere isdominated by a large-scale, low-frequency, quasi-horizontal velocity field. Charney–Drazin filtering(Charney and Drazin 1961) permits propagation of onlythe planetary-scale Rossby waves into the wintertimestratosphere. Of particular interest is the erosion of thepolar vortex as these planetary waves break (Juckesand McIntyre 1987), resulting in filamentation as ma-terial is ejected from the polar vortex into the “surfzone,” a region of strong mixing at midlatitudes (McIn-tyre and Palmer 1983; Shepherd 2000). This process offilamentation can be described in terms of chaotic ad-vection, a phenomenon wherein a quasi-regular veloc-ity field gives rise to irregular particle paths (Pierre-humbert 1991; Ngan and Shepherd 1999a; Joseph and

Legras 2002). Filamentation has been the subject ofmuch study with regard to the dynamics of a perturbedvortex (e.g., Polvani and Plumb 1992), in part becausesuch an “erosion–entrainment” process sets up condi-tions for chemical ozone depletion in the springtimepolar vortex.

By contrast, the mesosphere is believed to be distin-guished by ubiquitous gravity wave activity. Due to de-creasing density with increasing altitude, internal iner-tia–gravity waves grow in amplitude as they propagateupward into the mesospheric region. Although ad-equate global observational data is not available for themesosphere, owing to the stringent spatial and tempo-ral resolution requirements necessary to accurately cap-ture mesospheric winds, local measurements suggestthe ubiquitous presence of gravity waves (Fritts andAlexander 2003). Moreover, wind fields generatedfrom general circulation models (GCMs) suggest thatthe mesosphere is characterized by increasing spatialirregularity with altitude (Koshyk et al. 1999). As dem-onstrated by Shepherd et al. (2000), GCMs also exhibitan increased temporal variability in the mesosphererelative to the stratosphere, as is manifested, for ex-ample, in the sensitivity of contour advection calcula-tions to temporal resolution of the winds.

* Current affiliation: Centre for Earth Observation Science(CEOS), University of Manitoba, Winnipeg, Manitoba, Canada.

Corresponding author address: Dr. T. G. Shepherd, Dept. ofPhysics, University of Toronto, 60 St. George Street, Toronto,ON, M5S 1A7, Canada.E-mail: [email protected]

NOVEMBER 2005 L U K O V I C H A N D S H E P H E R D 3933

© 2005 American Meteorological Society

JAS3580

Differences in stratospheric and mesospheric dynam-ics are quantified, in part, by kinetic energy spatialwavenumber spectra and their slopes. In terms of anenergy spectrum with power-law scaling E(k) � k��,comparison of flow fields generated from differentGCMs demonstrates that the stratosphere is character-ized by a steep spectrum with � � 3, with the divergentcomponent much weaker than the rotational compo-nent, indicative of balanced (slow, vortical) motion;whereas the mesosphere is distinguished by a muchshallower spectrum at smaller spatial (higher wavenum-ber) scales with � � 3, with the divergent and rotationalcomponents of comparable amplitude, indicative of un-balanced (fast) motion (Koshyk et al. 1999). Althoughthese are only GCM results, they are consistent withaircraft measurements of shallow kinetic energy spectraat smaller spatial scales in the lower stratosphere (Bac-meister et al. 1996).

Noteworthy, and key to the present study, are analo-gous spectral slope properties for what are referred toas (spectrally) nonlocal and local dynamics (Shepherdet al. 2000). For nonlocal dynamics, tracer evolution ata given length scale is governed by velocity features atmuch longer length scales; for local dynamics, tracerevolution is governed by velocity features at compa-rable length scales. As outlined by Bennett (1984) fornondivergent (balanced) flow, for a kinetic energyspectrum described by E(k) � k��, nonlocal dynamicsare characterized by � � 3. For 1 � � � 3, tracerevolution is described by local dynamics, where nosingle time scale exists. For � � 1, transport is describedby diffusion.

Although the above categorization of spectral slopesserves as a means by which the nature of mixing withina given regime may be established, measurements fromatmospheric observations or laboratory experimentstend to be, instead, of structure functions and diffusivi-ties. The kinetic energy spectrum is then derived fromthese measured quantities. The structure function is de-fined for u(x), the Eulerian velocity in physical spaceaccording to (Babiano et al. 1985)

S�D� �12

|u�x D� � u�x�|2�, �1�

for x � (x, y) the position vector and D � (Dx, Dy) theseparation vector with magnitude D, and where aver-aging (denoted by the angle brackets) occurs over allEulerian coordinates. According to Babiano et al.(1985), passive tracer behavior is governed by nonlocaldynamics and thus by the largest spatial scales for � �

3. Nonlocal dynamics are then depicted by a structurefunction scaling law of the form

S�D� � D2. �2�

Local dynamics, for which 1 � � � 3, involve lengthscales comparable to the interparticle separation so thatk � D�1. The structure-function scaling law is then ofthe form

S�D� � D��1. �3�

For the diffusive regime, where � � 1, the structurefunction is constant with respect to the interparticleseparation.

The present work is motivated by the desire toachieve both a qualitative and a quantitative under-standing of the effect of an unbalanced componentupon the topological and statistical properties of theflow. Of central interest is the influence of the unbal-anced component of the velocity field upon the coher-ent features prevalent in a regime described by chaoticadvection. Pertinent questions to be asked in this inves-tigation are how is balanced flow modified with thepresence of an unbalanced component? More specifi-cally, how does the unbalanced component affect thevelocity field and its spatial gradients, and how are un-balanced dynamics reflected in zonal and meridionaldispersion statistics? In addition, what are the implica-tions of this modified flow topology for mixing? Thisstudy addresses these questions through the examina-tion of zonal and meridional dispersion statistics. Theformer provide a framework against which theoreticalarguments such as anomalous dispersion and Richard-son’s (1926) t3 law may be tested, in contrast to thelatter which, on the sphere, are affected by geometricalconstraints. While previous studies have examined (bal-anced) transport in the context of dispersion along andacross potential vorticity (PV) contours (LaCasce andSpeer 1999), the present investigation seeks to quantifyzonal and meridional dispersion in the stratosphericsurf zone, where PV coordinates are not appropriate.Louazel and Hua (2004) have examined the effects ofdivergent flow on PV evolution, although not on theevolution of an arbitrary tracer, as is considered in thepresent analysis. The present study seeks also to deter-mine whether diffusion is an appropriate descriptionfor meridional transport in the context of mesosphericdynamics. The implications for this investigation residein the additional question: Is diffusion an appropriateparameterization for transport by unbalanced flow?

To address these questions, we analyze two idealizedshallow-water numerical experiments representative ofstratospheric and mesospheric dynamics. Section 2 de-scribes the shallow-water model, the experimentalsetup used to establish a surrogate stratosphere andmesosphere, and the kinetic energy spectra obtained

3934 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62

from these experiments. Methods and diagnostics usedto analyze the experiments, including the generaliza-tion of the Hua–Klein criterion to divergent flow, areoutlined in section 3. Section 4 presents results fromthese experiments. Here it is demonstrated that small-scale imbalance significantly influences meridional and,to a lesser extent, zonal dispersion, as highlighted bytwo-particle (relative) dispersion statistics. Further, thephysical implications of unbalanced dynamics for mix-ing are examined in the context of structure functions,where we find that arguments from two-dimensional(2D) turbulence theory [(2) and (3)] appear insensitiveto the introduction of an unbalanced (fast) componentto the flow.

2. The shallow-water model: Balance andimbalance

Although stratospheric dynamics can be largely de-scribed in terms of an essentially nondivergent flow, thepresence of gravity waves in the mesosphere does notpermit such a description. A dynamical system of equa-tions that incorporates the slowly varying vortical mo-tion in addition to the rapidly varying gravity waves istherefore necessary. The rotating shallow-water equa-tions (SWE) provide such a framework. For a singlefluid layer with varying surface height � and bottomtopography located at z � �H, the rotating SWE aregiven by (e.g., Salmon 1998)

DuDt

fk u � �g��, �4�

�h

�t � · �uh� � 0, �5�

where u � (u, �) is the horizontal velocity, � � (�x, �y),f is the Coriolis parameter, g is the gravitational accel-eration, k is the unit vector in the vertical, D/Dt is theLagrangian derivative, and h � � H denotes thedepth of the fluid layer. Various investigations (e.g.,

Ngan and Shepherd 1999a,b) have used the SWE toanalyze transport in the stratosphere. Here we examinethe dynamics resulting from the inclusion of a divergentcomponent in order to analyze transport in a regimecharacteristic of mesospheric dynamics and contrastthis with the stratospheric case.

Analyzed in the present investigation are two shal-low-water numerical simulations representative ofstratospheric and mesospheric dynamics on a quasi-horizontal isentropic surface. The first, referred to asthe planetary-wave (PW) experiment, consists of plan-etary-wave forcing superimposed on a zonal shear flow,and is established to represent dynamics in the winter-time extratropical stratosphere. This setup replicatesthat of previous studies (e.g., Juckes and McIntyre1987; Ngan and Shepherd 1999a,b) and uses the sameheight and mean shear as in the studies of chaotic ad-vection of Ngan and Shepherd (1999a,b). Planetarywave forcing is intended to generate dynamics resultingfrom Rossby waves in the wintertime stratosphere,which are normally of zonal wave 1 or wave 2, and thusto reflect the mixing properties of a balanced flow field.The second experiment, referred to as the planetarywave/gravity wave (PW/GW) experiment, consists offorced gravity waves in addition to planetary waves su-perimposed on the basic state, and is established torepresent mesospheric dynamics (albeit in a highly ide-alized fashion). This is the novel regime and is of inter-est in itself as an example of strongly divergent flow. Itshould also be noted that the regime investigated in thepresent study is that of nonbreaking gravity waves,where the gravity waves provide a high-frequency,small-scale stochastic component to the horizontal ve-locity field, and particles are confined to isentropic sur-faces. For this purpose, a two-dimensional study is ap-propriate. Clearly, the study is not as relevant to thecase of breaking gravity waves, where three-dimensional effects would become important.

The additional forcing function for the PW/GW ex-periment in spectral space is described by

�GW�n, m� � ��0GW�n*�n* 1�

n�n 1� �1�6

exp i��n, m, t�, if n* n N

0, otherwise.�6�

Here � � gz is the geopotential of the bottom topog-raphy located at depth z � �H, N is the model trun-cation wavenumber, n is the spherical harmonic index,m is the zonal wavenumber, n* � 3 is the minimumtotal wavenumber forced for the GW case, and

�0GW � 1.5 10 m2 s�2 is the amplitude of the gravitywave forcing. Parameters chosen for the PW/GW ex-periment were found by trial and error to approxi-mately replicate the kinetic energy spectra computedby Koshyk et al. (1999), and analyzed by Shepherd et al.


(2000), on the isentropic surface 4000 K (middle meso-sphere). The phase �(n, m, t) is modified temporally viaa recursive filter scheme

�new�n, m, t� � �old r�1 � 2�1�2, �7�

where � defines the temporal autocorrelation � of theforcing according to the relation �/�t � 1/(1 � �), whichmeasures the memory time in terms of the time steps�t, and r is a random number found between 0 and 2�.Here � � 0.92, corresponding, for model time steps of�t � 15 min, to an e-folding time � of approximatelythree hours.

In the statistical analyses, a third experiment consist-ing of gravity waves with a temporally decorrelatedflow field (i.e., the recursive filter scheme implementedfor gravity wave forcing is replaced by a random tem-poral forcing) superimposed on the zonal shear flow,without the PW forcing, is used as a benchmark forBrownian motion, and is referred to as the gravitywave, random in time (GWRT) experiment. The PWand GWRT experiments represent two extremes of bal-anced and unbalanced flow.

For the current investigation, the model was run withlongitude–latitude spatial resolution of 145 72 points,and sampled every three hours. The geopotential andvelocity field are relaxed to the westerly jet (zonal shearflow) on a time scale of ten days. The runs extended fora model time of 70 days, in order to ensure that statis-tical equilibrium was attained. Subsequent analyses ne-glect the first 20 days of the run, unless otherwise noted,in order to avoid spurious effects resulting from theinitial implementation of the forcing mechanisms.

Shown in Fig. 1 are snapshots of the zonal wind fieldin the PW and PW/GW experiments illustrating, in ad-dition to the zonal jet, the presence of large-scale co-herent features in the PW experiment. In contrast,small-scale features are evident in the PW/GW experi-ment. Spatial and temporal autocorrelations (notshown) further characterize both the similarity and thedistinction between the simulated stratospheric andmesospheric regimes. In particular, similarity in the lon-gitudinal spatial correlation of the geopotential for thetwo experiments may be attributed to the large-scalefeatures that are set by the planetary-wave forcingmechanism. Differences in temporal behavior aremanifested in a much more rapid decorrelation time(on the order of an hour) in the total geopotential forthe PW/GW experiment than for the PW experiment.In the latter case, the total geopotential does not appearto decorrelate at all for the time interval examined(over four days for quarter-hour sampling from days 60

to 65), indicative of temporal periodicity in the PWfield. The time scale for the PW/GW experiment is thetime scale of the gravity waves themselves.

Figure 2 depicts the temporally averaged kinetic en-ergy spectra for the total, rotational and divergent com-ponents of the winds for the extratropical NorthernHemisphere. The total-energy power spectrum for thePW experiment exhibits a steep spectral slope, with therotational component dominating the divergent com-ponent. The peak at n � 4 (length scale �4000 km)indicates energy associated with the large-scale, low-frequency PWs. For the PW/GW experiment, the steepspectral slope at large spatial scales (small wavenum-ber) gives way to a shallower spectral slope at smallerscales, where the divergent component is comparableto (in fact, greater than) the rotational component. Thisis qualitatively consistent with the stratospheric andmesospheric kinetic energy spectra found by Koshyk etal. (1999) in GCMs.

FIG. 1. Instantaneous snapshots of the zonal wind field (m s�1)at t � 70 days for the (a) PW and (b) PW/GW experiment, withcontour intervals of 10 m s�1.


Time scales associated with the PW and PW/GW ex-periments are illustrated in the frequency power spec-tra for the divergence and the linearized potential vor-ticity � � h� f/h* for the PW and PW/GW experiments,where � � �x� � �yu denotes the relative vorticity, h*denotes the mean fluid depth of 8 km, and h� � h � h*,and are depicted in Fig. 3. Observed in the power spec-tra for the divergence is an accumulation of power athigh frequencies for the PW/GW experiment. Thepower spectra for the linearized potential vorticity in-dicate an accumulation of power at low frequencies forboth the PW and PW/GW experiments, but also signifi-cant power at high frequencies for the PW/GW experi-ment, suggesting a fast component to the linearized po-tential vorticity in this case.

3. Methods and diagnostics

a. Geometric analysis: The Hua–Klein criterion andpatchiness

To understand the nature of transport and mixing inthese experiments and to determine the influence of theunbalanced component to the flow on the velocity fieldand its gradients, we characterize geometric aspects ofthe velocity field using the Eulerian diagnostic of theHua–Klein criterion and the Lagrangian diagnostic ofpatchiness.

The behavior of passive tracer motion in two-dimensional flow may be elucidated by a partitioning ofthe flow into vorticity-dominated (elliptic) and strain-dominated (hyperbolic) components, known as the

FIG. 2. Time-averaged (20–70 days) kinetic energy spectra (m2 s�2) for the (a) PW, (b) PW/GW, and (c) GWRT experiments.Indicated in each panel are the total, rotational, and divergent flow components. In (a) the total and rotational overlap are shown.

FIG. 3. Power spectra of (a) the divergence � (s�2) and (b) a linearized and scaled potential vorticity � � h�f/h*

(s�2)for the PW (solid) and PW/GW (dashed) experiments.


Okubo–Weiss criterion (Okubo 1970; Weiss 1991). Thisis achieved through consideration of the eigenvalues ofthe Eulerian velocity gradient tensor, �u/�x, determinedfrom the linearized equations of relative motion

x ��u�x

· x, �8�

for x the (Eulerian) relative position vector, and wherethe dot denotes the partial time derivative, �/�t. Thereal component of the square of the eigenvalues of thevelocity gradient tensor then yields the expression forthe Okubo–Weiss criterion applicable to divergentflow,

QOW �14��2 S2 � �2 2��S2 � �2�, �9�

where

S2 � Sn2 Ss

2 �10�

for Sn � �xu � �y� the stretching rate, Ss � �x� �yu theshear component of the strain, � � �x� � �yu the rela-tive vorticity and � � �xu �y� the divergence (Proven-zale 1999). Regions with QOW � 0 correspond to strongdivergence or deformation. Regions with QOW � 0 in-dicate the dominance of elliptic fixed points.

However, the Okubo–Weiss criterion, in assumingthat velocity gradients vary slowly in time with respectto vorticity gradients, has been shown to hold only atthe center of vortex cores and near stagnation points(Basdevant and Philipovitch 1994; Hua and Klein1998). [Lagrangian versions of the Okubo–Weiss crite-rion, which monitor the finite-time hyperbolicity of atrajectory, relax the conditions of this assumption(Haller 2000; Lapeyre et al. 2001).] In general, the timerate of change in the velocity gradients must be consid-ered. Such is the premise upon which the Hua–Kleincriterion (Hua and Klein 1998) is founded. Particle mo-tion is described in terms of the Lagrangian accelera-tion

�L � x. �11�

The Hua–Klein criterion QHK is defined as the largestof the eigenvalues of the acceleration gradient tensorfor nondivergent flow. Generalization of QHK to en-compass divergent flow is determined by substituting(8) into (11), giving

.

�L � ��u�x� · x ��u

�x�2

· x. �12�

The eigenvalues of the acceleration gradient tensor inthe divergent case are hence expressed as

� �14�2� �2 S2 � �2�

�12�Ss

2 Sn2 � �2 2��SsSs SnSn ��

�2�S2 � �2�.

�13�

The diagnostic implemented to investigate the evolu-tion of the velocity gradients is defined by the largesteigenvalue, �, which we call QL. For nondivergentflow, QL � QHK. In the limit of slowly evolving fields,but nonvanishing divergence, QL � QOW. Large posi-tive values of QL correspond to strain-dominated re-gions and negative values to vorticity-dominated re-gions. It is this diagnostic that we use as the basis forour Eulerian geometric analysis.

Louazel and Hua (2004) recently developed a gener-alization of the Hua–Klein criterion to assess the role offast dynamics on vortex filamentation in a shallow-water model. Their diagnostic differs from ours in thattheir generalized Hua–Klein criterion is evaluated instrain coordinates and incorporates divergence in thematerial derivative of the normal and shear strain com-ponents. By contrast, our analysis explicitly computesthe divergence in the original (unrotated) coordinatesystem.

Geometrical structure may also be assessed using thediagnostic of patchiness or the representation of dis-tinct average velocities in the flow (Pasmanter 1988;Malhotra et al. 1998). For a given fluid particle trajec-tory, the Lagrangian mean velocity measured along theparticle path is expressed (in the notation of Malhotraet al. 1998) as

�x ��x�t� � x�0��

t, �y �

�y�t� � y�0��t

�14�

for an interval of time t. While the Hua–Klein criterionexamines kinematic features associated with the geo-metric properties of the flow, patchiness may be used toanalyze its dynamical properties, through investigationof the evolution of the average Lagrangian velocity. Ithas been shown (Malhotra et al. 1998) that patchiness isuseful for the detection of large-scale inhomogeneitiesin the flow.

b. Statistical analysis

In this investigation we seek also to determine theinfluence of unbalanced flow on zonal and meridionaldispersion statistics. Complementary to the geometrical


assessment performed using the Hua–Klein and patchi-ness diagnostics is a statistical assessment based onsingle-particle (absolute) and two-particle (relative)dispersion. Absolute dispersion monitors the transla-tion of a single particle (Taylor 1921) and is defined as(in the notation of Provenzale 1999)

A2 � |xi�t� � xi�0� � xi�t� � xi�0��|2�

�1N�i�1

N

|xi�t� � xi�0� � xi�t� � xi�0��|2, �15�

where xi(t) denotes the position of the ith particle attime t, and N is the number of particles in the ensemble.Angle brackets denote averages over the particle en-semble, and dispersion can be defined in both the zonaland meridional directions.

Of particular interest in describing the properties of aflow using single-particle dispersion is the scaling be-havior

A2 � t� �16�

when ! 1 and ! 2. Anomalous diffusion refers tothe phenomenon wherein long-time, large-scale tracertransport exhibits behavior that departs from the ex-pected diffusive limit � 1. Ballistic dispersion ( � 2)is a special case of superdiffusion ( � 1), which isdistinguished by a mean-square displacement thatgrows more rapidly than linearly with time and is asignature of long-range correlations in the Lagrangianvelocity, corresponding to large-scale (and low-frequency) inhomogeneities in the Eulerian velocityfield. In contrast, subdiffusive behavior ( � 1) is dis-tinguished by a mean-square displacement that growsmore slowly than linearly with time and is characteristicof trapping events, which would occur if particles fol-lowed regular orbits and experienced a coherent, tem-porally periodic velocity field.

It is also of interest to note the link between absolutedispersion and the Lagrangian velocity autocorrelation.In the case of statistically stationary flow, theLagrangian integral (decorrelation) time scale TL is de-fined as (Majda and Kramer 1999)

TL � �0

�

R�� d�, �17�

where

R�� V�t�V�t ��

V2�t��18�

is the Lagrangian autocorrelation, and V is the depar-ture of the Lagrangian velocity from the ensemble av-

erage. For sufficiently short times, t � TL, where t is theelapsed time, the particles have memory of their initialvelocities, and temporal dispersion is described by theballistic regime. By contrast, for very long times, t � TL

(assuming that TL � "), the particles lose memory oftheir initial velocities and follow independent randomwalks as in Brownian motion so that the diffusive limitis attained. When statistical stationarity cannot be as-sumed, (17) is also a function of time so that a temporalaverage of the ensemble is required. Nevertheless, aqualitative comparison between autocorrelation anddispersion behavior may still be used to compare thestatistical properties of balanced and unbalanced flow.

The statistical analyses presented in section 4 arebased on an ensemble of 21 824 particles, initiallyaligned along latitudes separated by 1.0° and with alongitudinal spacing of 0.5°. The particles are initiallyconfined to the domain bounded by latitudes 15.5° and52.5°. Tracer trajectories and calculations of zonal dis-persion allow for wrapping past 360° longitude. All ab-solute and relative dispersion probability distributionfunctions (PDFs) are normalized to have unit area. Theadvection is performed using a particle advectionscheme developed by K. Ngan (see Ngan and Shepherd1999a), with a time step of three hours, correspondingto the model wind sampling interval.

The PDF of particle displacements provides a mea-sure of the evolution of a passive scalar. Departuresfrom Gaussianity reflect the presence of long-rangecorrelations. The existence of long-range correlationsin particle transport is manifested in the PDF skewness

S �X3�

�3 , �19�

and the flatness

F �X4�

�4 , �20�

where # ��A2 is the standard deviation of the singleparticle displacement, and X � x(t) � x(0) � x(t) �x(0)�. Skewness is a measure of asymmetry in the PDFand is zero in the Gaussian limit; flatness monitors dis-tribution broadness, and F � 3 for Gaussian PDFs.Broader-than-Gaussian tails with F � 3 indicate long-range correlations in the Lagrangian velocity that, inturn, give rise to large particle displacements. As theadvected fluid particles encounter coherent structures,dispersion by Brownian motion, expected for homoge-neous turbulence, is replaced by anomalous transportevidenced in non-Gaussian PDFs.

While absolute dispersion monitors the translation ofa single particle, relative dispersion monitors the sepa-


ration between a pair of particles. The concept of rela-tive dispersion was introduced by Richardson (1926)and is defined as

D2�t; t0; D0� �1N�i�1

N

|xi�t� � xi1�t� � xi�t� � xi1�t��|2,

�21�

for D0 the initial separation between a pair of particlesxi and xi1, N the number of pairs with initial separationD0, and where angle brackets again denote an ensembleaverage. Here, as in (15), dispersion can be defined inboth the zonal and meridional directions. [It should benoted that in the relative dispersion analysis, the mag-nitude of separation between particle pairs is moni-tored so that xi(t) � xi1(t)� � 0, where angle bracketsdenote an ensemble average, in order to guaranteesymmetry in labeling.] The separation vector is herein-after denoted as D � (Dx, Dy) with magnitude D. Inmonitoring the relative motion between a pair of par-ticles, relative dispersion incorporates the effects ofvarying sized eddies upon a cluster of particles.

Relative dispersion monitors the straining mecha-nisms acting on scales comparable to the interparticleseparation and is thus dependent on velocity gradients,rather than expressly on the velocity field itself as issingle-particle dispersion (Batchelor 1951). This sug-gests that the correlations of the Lagrangian velocitygradients, instead of the Lagrangian velocities them-selves, provide a link between relative dispersion andtemporal autocorrelations. One would therefore expectthe Eulerian diagnostics of QOW and QL, which moni-tor the combined effect of velocity gradients, to be re-flected in the relative dispersion statistics. Moreover,the time evolution of the separation of a particle pairreflects the accelerating process of relative dispersion(Babiano et al. 1990). This is demonstrated by the dif-ferential equation for the separation (Babiano et al.1990)

d

dt � |D � D0|2

t � � |W�t, D0�|2 ��1t �0

t

�dW��

d�d��2

,

�22�

where W � (Wx, Wy) is the relative Lagrangian velocityassociated with the relative separation vector D. In par-ticular, the inclusion of the acceleration term in theexpression for the time evolution in the separation be-tween a pair of particles is reminiscent of the Hua–Klein criterion, and incorporates a sensitivity to unbal-anced motion. For the short-time limit t → 0, the zonalrelative acceleration is constant as the particles undergo

linear displacement so that the zonal relative dispersionsatisfies

Dx2 � D0x

2 2D0x · Wx�t��t 2sx�D0x�t2, �23�

where sx(D0x) � |Wx (D0x)|2�/2 is the second-order,one-dimensional structure function. In the long-timelimit t → ", relative dispersion is characterized by dif-fusion, as particle pairs lose memory of their origin, sothat D2 � t. Richardson’s (1926) t3 law holds for inter-mediate times under the assumption of statistical sta-tionarity in the Lagrangian relative accelerations (as isnoted in Zouari and Babiano 1994), and was initiallyderived by Richardson from observations under the as-sumption of a separation-dependent diffusivity.

c. Structure functions: Spectral nonlocality andlocality

The implications of modified flow topology resultingfrom unbalanced dynamics for mixing are addressedusing the concepts of spectral nonlocality and locality.Spectrally local and nonlocal properties of stratosphericand mesospheric dynamics, as determined from the PWand PW/GW experiments, respectively, may be quan-tified using the scaling laws derived from the structurefunctions outlined in the introduction.

The structure functions presented in section 4 arecomputed from the winds according to (1), for displace-ments in the zonal direction only. It is assumed that dueto similarities in the zonal and total dispersion resultingfrom the zonal shear flow, structure functions in thezonal direction would provide a valid interpretation ofspectrally nonlocal and local behavior for the total(zonal and meridional) flow. Averaging for this pur-pose occurs over time and over all longitudes at 45°N.

d. Eddy diffusivities

Eddy diffusivities can be used in the development oftracer mixing parameterization schemes that model un-balanced flow, as determined from the aforementionedstatistical analysis. Passive tracer behavior for a tracer Tis governed by the advection–diffusion equation

�T ��t

V · �T � � � · �K · �T ��, �24�

where K refers to an effective eddy diffusivity, or thesum of the molecular diffusivity k and a turbulent eddydiffusivity KT resulting from fluctuations in the advect-ing velocity v � V v �, for V the mean advectivevelocity and v � the random, turbulent fluctuations. Itshould be noted that (24) is not a closed equation: theprocess of averaging introduces a new variable KT, viav � · �T � � �� · (KT · �T �) (Majda and Kramer 1999).


It is the closure problem that is responsible for theambiguity inherent in defining a generalized advection–diffusion equation and, as a consequence, an eddy dif-fusivity.

Pertinent to the present discussion is the relationshipbetween the eddy diffusivity and single-particle disper-sion (Batchelor 1949). The absolute eddy diffusivity isdefined such that (Babiano et al. 1990)

KA �12

dA2

dt, �25�

where A2 denotes the absolute dispersion as defined in(15).

Richardson (1926) addressed the question of thepresence of a wide range of eddy diffusivities from at-mospheric measurements. Richardson’s study was mo-tivated by a desire to determine a generalized eddydiffusivity, independent of particle location, but depen-dent on particle separation. Richardson noted that aconstant diffusivity provides an incomplete descriptionof atmospheric dispersion in the presence of eddies, asis evidenced in the expression for the relative diffusivityKR, or diffusivity of neighbors (Richardson 1926)

KR � �D4�3, �26�

where $ is a constant independent of relative separa-tion, obtained from empirical arguments and observa-tions.

The analytic expression for the relative diffusivity, ortime evolution in the separation variance, underlinesthe role of relative dispersion in addressing the range ofeddy diffusivity values in the atmosphere. The relativediffusivity is defined, in a manner akin to the definitionfor the absolute eddy diffusivity (25), as (Babiano et al.1990)

KR�t, D0� �12

dD · D�dt

�12

dD2�t, D0�

dt, �27�

for D2 � D · D�, and where the arguments D0 and thave been introduced to emphasize the dependence ofthe eddy diffusivity on the initial interparticle separa-tion D0 and on time t. Thus, in contrast to the absolutediffusivity KA, the relative diffusivity need not be con-stant in time in order to be well defined. In fact, thesignificance of the relative diffusivity rests on its abilityto demonstrate that the rate of diffusion increases withinterparticle separation (Richardson 1926).

It is also possible to determine an (instantaneous)relative diffusivity, defined, in contrast to (27), as

Ki �12 �� d

dtD · D�21�2

, �28�

to be computed from the expression (Babiano et al.1985)

Ki � DS�1�2�D�, �29�

where here the longitudinal structure function S||(D) isnow dependent on D the interparticle separation basedon a conditional ensemble averaged where |D| � D,independent of the initial interparticle separation. Thisallows for the diffusivity to be specified in a diffusionparameterization scheme based on an appropriatemodel grid size.

4. Numerical results

a. Geometric analysis

The role of unbalanced dynamics in modifying flowtopology is shown in the Okubo–Weiss (QOW) and gen-eralized Hua–Klein (QL) fields for the PW and PW/GW experiments (Fig. 4). Noteworthy is the erosion oflarge-scale spatial gradients with the inclusion of anunbalanced component to the flow, evidenced in thedestruction of the strain-dominated region for the PW/GW as compared to the PW case. This indicates thatvelocity gradients are indeed modified by fast dynam-ics.

For the PW experiment QOW is rather similar(though certainly not identical) to QL. This suggeststhat the velocity gradients are slowly varying in timewith respect to the vorticity gradients, which in turnindicates that the changes in the rotational componentof the flow, associated with balanced (slow) dynamics,dominate. Moreover, the similarity in QOW and QL inspite of the time-dependent terms evident in the latterimplies the existence of a single advective time scaleimposed by the planetary waves superposed on the ba-sic-state shear. This is suggestive of nonlocal dynamics,as described in the introduction.

A much more marked contrast between the QOW andQL fields is found for the PW/GW experiment, as dem-onstrated in the order of magnitude increase in QL rela-tive to QOW, and as is to be expected owing to the largeunbalanced component associated with the high-frequency gravity waves. Hence the strain and diver-gent components evolve more rapidly than for balancedflow so that the significant gradients in vorticity foundfor the PW experiment are replaced by comparably andmore rapidly evolving gradients in the strain and diver-gent components. This dissimilarity highlights the exis-tence of a rapidly evolving (gravity wave) field thatintroduces numerous time scales for the PW/GW caseand implies spectrally local dynamics.


Shown in Fig. 5 are the patchiness plots for the zonalwinds in the PW and PW/GW experiments at t � 0, 10,and 20 days. The initial distribution (Figs. 5a and 5d)corresponds to the Eulerian zonal velocity field. Evi-dent in Fig. 5a is the wave-1 cat’s eye structure resultingfrom the PW forcing. Regions of maximal patchiness att � 0 coincide with maximum instantaneous zonal ve-locities. The dominance of the shear flow in the zonaldirection gives rise to large zonal particle displace-ments, as shown by the maximum values at the core ofthe zonal jet. Moreover, dominant coherent featuresremain largely intact for the PW/GW experiment, as isdemonstrated in Figs. 5d–f. In fact, there is an organi-zation and intensification of the structure as time pro-ceeds and the integrated effect of the small-scale fea-tures averages out, leaving the effect of the low-frequency, large-scale structure.

The variability found at the edge of the eastward jetfor the PW/GW experiment suggests that some mixingoccurs at the periphery of the jet because trajectorieswith different Lagrangian histories are contributing tothe structure. Enhanced transport into and out of thecat’s eye structure is apparent at t � 0, where the cat’s

eye takes on a more diffusive character for the PW/GWexperiment and, as previously noted, at later time or-ganizes to leave a large-scale, low-frequency structuresimilar to the PW case. The erosion of patches is con-sistent with the behavior predicted by Pasmanter (1988)in considering the effects of turbulent diffusion in thepresence of coherent structures, and with the effects ofinclusion of noise as examined by Malhotra et al.(1998). Larger patches are relatively insensitive to thespatially random and high-frequency fluctuations: it isonly the smaller patches that disappear. While the vari-ability in patchiness geometry with the gravity wavesmight be thought to imply increased transport and per-haps mixing, the persistence of the coherent features inthe zonal direction discards such a proposition. Disper-sion in the direction of the zonal shear flow would ap-pear to be largely unaffected by the presence of thegravity waves.

This behavior is in contrast to transport transverse tothe zonal shear flow. Whereas transport in the zonaldirection is governed by the combined effects of thebasic-state shear flow, planetary-wave and gravity waveforcing, transport in the meridional direction is gov-

FIG. 4. Grayscale plots determined from the Okubo–Weiss criterion, QOW, for the (a) PW and (c) PW/GWexperiments, and from the generalized Hua–Klein criterion with a divergent component, QL, for the (b) PW and(d) PW/GW experiments, all at t � 45 days. Units are in s�2. Note the difference in scale between (c) and (d).


erned only by the planetary wave and gravity waveforcing. This is emphasized in the patchiness plots formeridional winds. Figure 6 shows that the initial patchi-ness plot for the PW/GW experiment is more diffusivein character than at t � 10 days. Smaller patches mergeto produce the larger patch at midlatitudes, which be-gins to disintegrate at t � 20 days. We speculate that the

introduction of an initially random field to the meridi-onal flow generates motion akin to Brownian motion: ifthe gravity waves give a random walk on top of the(spatially smooth) large-scale velocity field, then theireffect on �y rapidly goes like �t, and hence becomesnegligible compared to the contribution to �y from thelarge-scale motion, which goes like t. Consequently, the

FIG. 5. Patchiness plots [as a function of x(0)] for zonal winds (normalized) for the PW experiment at (a) t � 0days, (b) t � 10 days, and (c) t � 20 days, and for the PW/GW experiment at (d) t � 0 days, (e) t � 10 days, and(f) t � 20 days. Maximal (minimal) patchiness denoted by light (dark) shading.


patchiness “organizes” until the large-scale motion it-self breaks up (after 20 days). Noteworthy also is en-hanced variability in the surf zone for the PW/GW ex-periment relative to the PW experiment, demonstratingan increased possibility for particles to escape the mid-latitude region of strong mixing, and reminiscent of thesignificant erosion of the velocity gradients observed

for both the QOW

and QL fields in Fig. 4. Unbalanceddynamics, as manifested in gravity wave activity, aretherefore shown to provide a significant contribution todispersion in the meridional direction, given the ab-sence of mean meridional flow. These patchiness re-sults are further illustrated in the single-particle disper-sion results.

FIG. 6. Patchiness plots [as a function of x(0)] for meridional zonal winds (normalized) generated for the PWexperiment at (a) t � 0 days, (b) t � 10 days, and (c) t � 20 days, and for the PW/GW experiment at (d) t � 0 days,(e) t � 10 days, and (f) t � 20 days. Shading as in Fig. 5.


b. Statistical analysis

The effect of unbalanced dynamics on zonal and me-ridional dispersion statistics is captured in the en-semble-averaged Lagrangian autocorrelations for thezonal and meridional winds, computed for the PW, PW/GW, and GWRT experiments (Fig. 7). An ensemble of700 particles was used in contrast to the 21 824 particlesused in other statistical analyses performed here, due tomemory constraints. Evident in Fig. 7a is the absence ofa decorrelation time for the zonal wind: the presence ofthe zonal jet in the form of a basic-state shear flow es-tablishes long-range correlations in the flow in all threeexperiments. Anticorrelation in the winds over longtime scales may be a consequence of meridional par-ticle migration, or a signature of oscillatory behavior.The experiments would need to be run longer to con-firm the latter. While one cannot accurately estimateR(�) because of nonstationarity, one would neverthe-less expect, from the long-range correlations estab-lished by the zonal shear flow in all three experiments,zonal dispersion to be characterized by superdiffusion.

Indeed, ballistic behavior is observed in the en-semble-averaged absolute zonal dispersion for the PW,PW/GW, and GWRT experiments, depicted in Fig. 8.The similarity in the behavior of the three experimentssuggests that single-particle dispersion is insensitive tothe behavior of small-scale eddies. Neither the smaller-scale gravity waves resulting from the forcing at higherwavenumbers, or even the planetary wave, are reflectedin the absolute zonal dispersion. Rather, the large-scalesweeping effects exerted by the westerly jet dominate.These results are consistent with results found for hori-

zontal transport in a meandering jet (Samelson 1992). Itis also of interest to note that for mean-shear-controlleddispersion, the relationship A2 � �yu2t2 is satisfied inthe case of zonal dispersion. In particular, for a typicalvalue of �2.0 10�11 s�2 for QOW (see the rotation-dominated region in Fig. 4a near 50°N—as is evident inthe large gradients across the zonal shear flow shownin Fig. 5, the vorticity is dominated by �yu), �yu �

FIG. 7. Ensemble-averaged Lagrangian autocorrelations for (a) zonal and (b) meridional winds, averaged over all longitudes(15.5°–52.5°N). The inset in (b) shows a blow up of the PW/GW and GWRT curves for the first few days.

FIG. 8. Ensemble-averaged zonal absolute dispersion for thePW, PW/GW, and GWRT experiments. This and subsequentcomputations are performed using an ensemble of 21 824 par-ticles. Note that the offset at small t is due to the fact that A2 � 0at t � 0 so that, for the log–log plot, the initial dispersion is att � 3 h (advection time step).


QOWa2cos2(y)� 400 m2 s�2, which is of the same order ofmagnitude as the A2/t2 relation determined from Fig. 8.

Figure 7b depicts the temporal autocorrelations as-sociated with the meridional winds for the PW, PW/GW, and GWRT experiments. The distinction betweenthe PW and PW/GW experiments is evident in themore rapid decay of the Lagrangian correlation func-tion for the PW/GW experiment: the presence of a fi-nite Lagrangian time scale implies diffusive behavior inthe meridional direction in the long-time limit. Thepresence of negative correlations, as in the case of theautocorrelations for the zonal winds, may be an artifactof oscillatory behavior induced by planetary waves.However, as previously noted, longer experimentalruns would be required to verify this hypothesis.

The connection between Lagrangian autocorrelationbehavior and single-particle dispersion (Taylor 1921) asdescribed in section 3b is evidenced in the ensemble-averaged absolute meridional dispersion shown in Fig.9. In particular, the rapid decay in the autocorrelation isreflected in diffusive behavior in the latter stages of thePW/GW and GWRT experiments. A ballistic regime ( � 2) extends to t � 3 days for the PW and PW/GWexperiments, indicating that particles experience unidi-rectional motion until this time. Subdiffusive ( � 1)behavior, characteristic of trapping such as observed inLevy flights (Weeks et al. 1996), is apparent after t � 4days, when the absolute dispersion has a value of �3 105 km2, corresponding to an rms absolute displace-ment of approximately 500 km. Differences betweenthe PW and PW/GW experiments become apparentonly after t � 20 days. While dispersion in the PWexperiment plateaus, dispersion in the PW/GW experi-ment continues to evolve subdiffusively, thus demon-strating the ability of gravity waves to enhance the dis-placement of individual particles. More marked is thecontrast with the GWRT experiment: in that case, dif-fusive ( � 1) behavior is exhibited for most of thetemporal range considered. This is a consequence ofthe absence of a planetary wave: no coherent featuresare present in the meridional direction for this spatiallyand temporally decorrelated flow. Also, as demon-strated in the patchiness plots presented in Fig. 6, somestructure is apparent for the meridional component ofthe flow, although less dominant than for the zonalcomponent. These features are even less distinctive forthe PW/GW experiment. One can imagine, in a manneranalogous to the physical picture presented by Pasman-ter (1988), that both chaotic and regular particle trajec-tories exist in the meridional direction, so subdiffusiverather than superdiffusive behavior might be expected.This is, indeed, found to be the case for single-particlemeridional dispersion. Thus in contrast to zonal abso-

lute dispersion, differences between the stratosphericand mesospheric regimes are apparent in the meridi-onal absolute dispersion.

The time evolution of the zonal-displacement PDFs,expressed in terms of �x � x(t) � x(0), (where the PDFsare normalized to have area unity with a bin size of2200 km), is shown in Fig. 10. These PDFs exhibit longtails illustrative of eastward advection by the zonalshear flow in all three experiments. Translation of par-ticles occurs via large-scale advection, regardless of thepresence or absence of gravity waves. Comparison ofthe PDFs for the PW and PW/GW experiments dem-onstrates slightly enhanced zonal dispersion for thePW/GW experiment. A compressed PDF for theGWRT experiment suggests that gravity waves, in theabsence of coherent features established by the PWforcing, smear the wide range of sweeping velocities inthe zonal direction. This may be explained in terms ofTaylor’s (1953) theory of shear-induced dispersionwhere it was found that molecular diffusion acts to in-hibit along-shear transport for a solute of uniform con-centration. Strong GWRT forcing prevents particlesfrom remaining in the center of the westerly jet longenough to move a large distance zonally. Skewness andflatness (not shown) reveal non-Gaussian behavior, asis to be expected in the presence of a zonal shear flowand has been noted in previous stratospheric transportstudies (Seo and Bowman 2000).

The effect of unbalanced dynamics on meridionaldispersion is demonstrated in the PDFs for meridional

FIG. 9. Ensemble-averaged meridional absolute dispersion forthe PW (solid), PW/GW (dashed), and GWRT (dotted) experi-ments. Note the offset at small t for the same reason as describedin Fig. 8.


absolute displacement expressed in terms of �y � y(t)� y(0), and depicted in Fig. 11. Here, the role of thegravity waves in establishing a more diffusive regime isevidenced in the more Gaussian PDFs for the PW/GW

experiment relative to the PW experiment. While theskewness (not shown) is essentially zero for all threeexperiments—the flow is symmetric, that is, there is nopreferred direction of displacement in the absence of a

FIG. 10. Probability distribution functions for zonal absolutedisplacement at different instants in time for the (a) PW, (b)PW/GW, and (c) GWRT experiments.

FIG. 11. As in Fig. 10 but for the meridional absolutedisplacement.


basic-state shear flow in the meridional direction—Fig.12 shows that the flatness is close to F � 3 for thePW/GW and GWRT experiments, both of which aresignatures of the Gaussian behavior that characterizes adiffusive regime. Noteworthy also from the absolutedisplacement PDFs is the existence of exponential tailsfor the PW experiment. Often cited as a signature oflarge-scale intermittency (Majda and Kramer 1999), ex-ponential as opposed to Gaussian decay is deemed areflection of large fluctuations in passive tracer behav-ior. In the present case, we speculate that these fluc-tuations may be attributed to intermittency associatedwith nonlocal dynamics. Previous stratospheric analy-ses (Sparling and Bacmeister 2001; Majda and Kramer1999) have also found a direct link between scale-depen-dent exponential functions and increment PDFs, whichrepresent intermittency due to a separation in scalesbetween the advecting wind field and the passive tracerfield or, equivalently, nonlocal dynamics. The resultsfor the PW/GW and GWRT experiments are, in con-trast, reminiscent of the behavior predicted by the cen-tral limit theorem, namely the emergence of a Gaussiandistribution in the long-time limit (t � TL), for TL theLagrangian integral time scale, and thus local dynamics.

A central feature to be noted from the zonal relativedispersion statistics is the difference between resultsobtained for the PW and PW/GW experiments, in con-trast to zonal single-particle dispersion statistics. Shownin Fig. 13 are the Lagrangian autocorrelations for thelatitudinal and longitudinal gradients in the zonal (�yuand �xu, respectively) and meridional (�y� and �x�, re-spectively) winds. The meridional wind gradients areincluded in the present analysis since consideration ofvelocity gradients and hence vorticity, strain, and diver-gence in the context of relative dispersion requires con-sideration of the gradients in u and �, a feature exem-plified in the Hua–Klein diagnostic. Figure 13 indicatesthat the autocorrelations for the PW/GW and GWRTexperiments decay much more rapidly than those forthe PW experiment. Negative correlations may be asignature of oscillations in the PW experiment, or aconsequence of large positive latitudinal gradients ofthe zonal winds (evident in Figs. 2a and 5a). The largestspatial gradients in the Lagrangian velocity are found atthe edge of the zonal jet (see, e.g., the patchiness plots),hence in the latitudinal gradient of the zonal wind �yu.The slow decay in the PW experiment for �yu, which isthe dominant term for the vorticity � � �x� � �yu,suggests that the PW experiment evolves on very slowtime scales. By contrast, the rapid decorrelation in both�xu and �y�, which comprise the divergence � � �xu �y�, for the PW/GW experiment suggests that the un-balanced component dominates for t � 3 days. Studies

by Babiano et al. (1990) have demonstrated an inverserelationship between the characteristic time scale de-noting the transition from short [(23)] to intermediatetimes and velocity gradients. The existence of such atime scale (�3 days) for the PW/GW experiment sug-gests that the intermediate regime characterized by Ri-chardson’s (1926) t3 law will be attained more rapidly inthe presence of a divergent component (as noted insection 3c) than for the PW experiment.

The Lagrangian autocorrelation behavior is eluci-dated in the ensemble-averaged relative zonal disper-sion D2 versus time, depicted in Fig. 14. The dominanceof advection associated with a zonal shear flow is dem-onstrated in the longer duration of linear relative dis-persion for the PW experiment: linear dispersion char-acterizes short-time behavior in each of the three ex-periments due to the nonzero mean advective termW(t)� relative to the quadratic term in (23). Evident inparticular is the more rapid approach to Richardson’s(1926) t3 intermediate regime for the PW/GW com-pared to the PW experiment, and for the GWRT com-pared to the PW/GW experiment. The temporal vari-ability associated with the gravity waves enhances sepa-ration between particle pairs, thus destroying memoryof initial particle separation more rapidly than for thePW experiment. In particular, a pair of particles ini-tially trapped within a given region may be allowed toescape. This has important implications for regionssuch as the surf zone and the edge of the polar vortex inthe middle atmosphere: the introduction of gravitywaves allows particles trapped within the core of thecat’s eye structure to permeate surrounding regions,

FIG. 12. Flatness of the meridional absolute displacement PDFfor the PW, PW/GW, and GWRT experiments.


while particles found near the edge of the polar vortexmay wander into midlatitude regions. This mechanismis evident in the zonal relative dispersion statistics: thelinear behavior depicted in (23) is replaced more rap-idly by Richardson’s t3 regime for the PW/GW than forthe PW experiment, underlining the role of the diver-gent component in eroding coherent structures estab-lished by planetary waves. Such behavior is furtherdemonstrated in meridional relative dispersion statis-tics, where the trapping mechanism established by PWsin the meridional direction is perturbed by the GWs,allowing particles to escape the surf zone, as is dis-cussed below. Geometrical considerations have alreadyindicated that gravity waves destroy the coherence ofthe stirring mechanisms present in the PW case (see,e.g., Fig. 4). As such, gravity waves may be viewed as acatalyst for mixing, and relative dispersion the tool withwhich the nature of this mixing is observed.

Figure 15 depicts the ensemble-averaged meridionalrelative dispersion. Meridional dispersion is character-ized by a quadratic (t2) regime for short times (on theorder of hours) in the absence of a mean advective term[cf. (23)], and diffusive behavior at intermediate times,for the PW, PW/GW, and GWRT experiments. It takeslonger for the diffusive (t) limit to be attained for theGWRT experiment most likely due to the absence of acoherent PW structure (the surf zone), within whichspatial homogeneity is more rapidly attained. In theabsence of planetary waves such diffusive behavior per-sists for long times, as is demonstrated in the GWRTexperiment. However, the introduction of organizedstructure to the flow with the planetary wave forcingand the suppression of the gravity waves results in trap-ping phenomena in the PW experiment, as shown in theplateau corresponding to a meridional trapping regionof width �1000 km or �10° latitude. This value is com-

FIG. 13. Lagrangian autocorrelations for (a) �xu, (b) �yu, (c) �x�, and (d) �y�, namely, the velocity gradients.


parable to the meridional width of the region of strongvorticity at midlatitudes shown by the Okubo–Weissand Hua–Klein criteria in Fig. 4. As previously noted,the combined effect of planetary and gravity waveserodes this inhomogeneity, as shown in the continuedincrease in the meridional relative dispersion D2 after t� 20 days for the PW/GW experiment, yet the effect ofthe planetary waves is still reflected in the subdiffusivebehavior for t � 4 days, in striking contrast to theGWRT experiment. Thus, only after t � 10 days do

particle pairs lose memory of their initial separation[the elapsed time will differ for different D0; see, e.g.,(23)], at which time the effects of unbalanced dynamicsbecome apparent for the PW/GW experiment. Thelong-time meridional relative dispersion results areconsistent with those found from the meridional abso-lute dispersion analysis. Both emphasize the emergenceof diffusive behavior in the presence of an unbalancedcomponent to the flow.

As with single-particle statistics, Taylor’s (1953)theory of shear-induced dispersion is relevant to thecase of zonal relative dispersion, in the sense that asolute initially concentrated at a single point willspread. Connected also to Taylor’s theory of shear-induced dispersion is the phenomenon wherein meridi-onal dispersion enhances zonal dispersion. The meridi-onal separation PDFs are shown in Fig. 16. The PDFsfor the PW experiment vary little in time between 30and 50 days, which reflects the subdiffusive behaviorobserved in Fig. 15. The PDFs for the PW/GW andGWRT experiments reveal that unbalanced dynamicsresult in enhanced separation between particle pairs—the introduction of a fast component increases therange of eddies to which particle pairs are subjected,allowing particles to travel beyond the confines of thesurf zone. In the case of zonal relative dispersion,shown in Fig. 17, the most narrow distribution is thatassociated with the PW experiment since the separationbetween a pair of particles is monitored, rather than theindividual particle displacement. With the introductionof gravity waves, separation is enhanced in the zonaldirection, thus demonstrating the ability of meridionaldiffusive mechanisms to enhance zonal dispersion of aparticle cluster. The stretched tails evident in the PW/GW case signal large particle displacements induced bythe advection by the basic-state shear flow and plan-etary wave, combined with enhanced local separationresulting from diffusive processes. The difference inscales between Figs. 10 and 17 should be noted. Rela-tive dispersion analysis indicates that particles—withinthe solute, for Taylor’s theory, and tracer particles forthe present problem—are allowed to separate moreeasily with a rapidly fluctuating component to the flow.

c. Structure functions: Spectral nonlocality andlocality

The implications of a modified flow topology for mix-ing, and subsequent determination of an appropriateeddy diffusivity are illustrated in structure functionscaling laws. Shown in Fig. 18a are the structure func-tions associated with the PW, PW/GW, and GWRTexperiments computed at 45°N and averaged over 50days. The structure functions at other latitudes are simi-FIG. 15. As in Fig. 14 but for the meridional relative dispersion.

FIG. 14. Ensemble-averaged zonal relative dispersion for thePW, PW/GW, and GWRT experiments, represented by solid,dashed, and dotted lines, respectively.


lar. The scaling laws appropriate for nonlocal and localdynamics as derived for two-dimensional turbulence(Bennett 1984; Babiano et al. 1985) appear to beobeyed for each of the three experiments in that the

structure function results of Fig. 18b coincide with scal-ing laws established for the KE spectra shown in Fig.18b. That is, for the case of PW forcing where nonlocalinteractions are expected to dominate, a D2 scaling law

FIG. 16. PDF of the meridional separation at different instantsin time for the (a) PW, (b) PW/GW, and (c) GWRT experiments.

FIG. 17. As in Fig. 16 but of the zonal separation.


is obtained at all spatial scales in accordance with (2).However, with the presence of the smaller-scale eddiesresulting from the GW forcing in the PW/GW experi-ment, the structure function approaches an intermedi-ate scaling law of S(D) � D2/3. According to (3), thiscorresponds to E(k) � k�5/3, which roughly approxi-mates the kinetic energy spectrum achieved in this ex-periment (Fig. 18b). By contrast, the structure functionapproaches a constant for the GWRT experiment, as isto be expected for the diffusive regime established bythe spatially random forcing. Figure 18 suggests that thescaling laws derived for 2D turbulence do indeed holdfor unbalanced flow. That is, the structure-functionscaling laws appear to be insensitive to the presence ofa strong divergent component to the flow.

The insensitivity of 2D turbulence arguments to theintroduction of a divergent component suggests thatphysical parameters such as the instantaneous eddy dif-fusivity defined in (29) may be derived from structurefunction scaling laws, regardless of whether the flow isgoverned by unbalanced or balanced dynamics.

d. Eddy diffusivities

Both the absolute and relative dispersion statisticsdemonstrate that the PW/GW experiment exhibits be-havior similar to the PW experiment at short timescales and to the GWRT experiment over longer timescales. Evidence of Gaussian statistics in the �t rmsdisplacement, skewness S � 0 and flatness F � 3 for theabsolute dispersion analysis suggests that a constant dif-fusivity provides a plausible characterization of trans-port in the meridional direction (Fig. 9) for the PW/GWand particularly the GWRT experiment [see, e.g., (25)].Enhanced separation in the meridional separationPDFs in the presence of an unbalanced component tothe flow further supports a diffusive description fortransport in unbalanced flow. The apparent insensitiv-ity of the structure-function scaling laws derived for 2Dturbulence to the presence of an unbalanced compo-nent to the flow, as demonstrated in the structure func-tion result S � D2/3 (see Fig. 18), suggests that unbal-anced dynamics may be represented by an instanta-neous separation-dependent diffusivity, defined inaccordance with the model spatial resolution [see (29)].Noteworthy also is the fact that the results from thestructure function analysis for the PW/GW experiment(S � D2/3), corresponding to local dynamics, yield aninstantaneous relative eddy diffusivity that is consistent[see (26)] with Richardson’s (1926) predicted D4/3 scal-ing.

5. Discussion

The present investigation has sought to elucidate theinfluence of an unbalanced, divergent component tothe flow in the context of zonal and meridional adia-batic transport in an idealized stratosphere and meso-sphere. Geometrical and statistical analyses have dem-onstrated that the introduction of a strong unbalancedcomponent to the flow with the inclusion of gravitywaves, as manifested in shallow slopes of kinetic energyspectra, does indeed have important implications fortransport and mixing.

Geometrical considerations demonstrate that veloc-ity gradients are strongly influenced by the presence ofan unbalanced component to the flow. There is an ero-sion of velocity-gradient structure in the Hua–Klein di-

FIG. 18. Temporally averaged structure function at 45°N for thePW, PW/GW, and GWRT experiments. Indicated also are theslopes corresponding to n � 2 and n � 2/3. (b) Correspondingkinetic energy spectra (m2 s�2) with reference slopes indicated.


agnostic in the PW/GW experiment relative to the PWexperiment. Zonal patchiness plots exhibit a persis-tence in coherent spatial structure owing to the domi-nant role played by the basic-state shear in the compe-tition between the basic-state shear, PWs, and GWs. Bycontrast, meridional patchiness plots exhibit enhancedvariability in the competition between PWs and GWs,and an erosion of spatial structure in the surf zone.Thus unbalanced dynamics are shown to induce en-hanced dispersion in the meridional direction, in theabsence of mean meridional flow.

Statistical analyses reinforce the geometrical results.The survival of coherent features exhibited by patchi-ness is confirmed by anomalous diffusion and non-Gaussian PDFs obtained in single-particle zonal disper-sion analyses. The effect of unbalanced dynamics onzonal transport is reflected by relative rather than ab-solute dispersion statistics, in particular in the morerapid approach to Richardson’s (1926) t3 regime for thePW/GW experiment than for the PW experiment.Separation PDFs for the PW/GW and GWRT experi-ments demonstrate also that a rapidly fluctuating com-ponent to the flow enhances along-shear transport andzonal spread in a particle cluster, which is consistentwith Taylor’s (1953) theory of shear-induced disper-sion. By contrast, both the single- and two-particle sta-tistical analyses provide further evidence of more dif-fusive behavior in the meridional direction resultingfrom the inclusion of an unbalanced component to theflow. The plateau attained in meridional dispersion sta-tistics for the PW experiment is replaced by subdiffu-sive behavior for the PW/GW experiment as the gravitywaves act to erode the coherent surf-zone structure im-posed by the planetary wave. A purely diffusive regimeis established in the absence of such coherent features,as is demonstrated by the results for the GWRT experi-ment. The establishment of a more diffusive regime inthe presence of unbalanced motion is demonstratedalso in the appearance of Gaussian statistics for meridi-onal absolute and relative dispersion PDFs associatedwith the PW/GW and GWRT experiments.

Spectral locality and nonlocality for the stratospheric(PW) and mesospheric (PW/GW) regimes were ex-plored using structure function scaling laws. Large-scale sweeping effects indicative of stirring are repre-sented by the nonlocal behavior depicted in the D2 scal-ing for the structure function, in a manner consistentwith the balanced dynamics observed in k�3 scaling forthe kinetic energy spectra. The homogenization ofsmall-scale structure, a signature of mixing, is repre-sented by the local dynamics characterized by the D2/3

structure function scaling for a corresponding kineticenergy spectrum with �k�5/3 scaling indicative of un-

balanced dynamics. These results further suggest thatmeridional transport in the mesosphere may be mod-eled as a diffusive process, and quantified by an instan-taneous eddy diffusivity that can be determined fromstructure-function scaling laws.

Acknowledgments. This paper is based on the firstauthor’s Ph.D. thesis at the University of Toronto, dur-ing which time support was provided by fellowshipsfrom the Natural Sciences and Engineering ResearchCouncil of Canada, OGSST, the Department of Physicsat the University of Toronto, and the Sumner Founda-tion. Particle advection studies were performed usingcode developed by K. Ngan. The numerical code usedto compute the patchiness plots was provided by theControl and Dynamical Systems division of the Califor-nia Institute of Technology at the “Lagrangian Trans-port, Stirring and Mixing in Geophysical Flows” sum-mer school held in August 1999.

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