Lagrangian dynamics in stochastic inertia-gravity waveswtang/files/TTA10.pdfLagrangian dynamics in stochastic inertia-gravity waves Wenbo Tang, Jesse E. Taylor, and Alex Mahalov School

Lagrangian dynamics in stochastic inertia-gravity wavesWenbo Tang, Jesse E. Taylor, and Alex MahalovSchool of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287, USA

�Received 20 April 2010; accepted 21 October 2010; published online 2 December 2010�

For an idealized inertia-gravity wave, the Stokes drift, defined as the difference in end positions ofa fluid parcel as derived in the Lagrangian and Eulerian coordinates, is exactly zero after one wavecycle in a deterministic flow. When stochastic effects are incorporated into the model, nonlinearityin the velocity field changes the statistical properties. Better understanding of the statistics of apassive tracer, such as the mean drift and higher order moments, leads to more accurate predictionsof the pattern of Lagrangian mixing in a realistic environment. In this paper, we consider theinertia-gravity wave equation perturbed by white noise and solve the Fokker–Planck equation tostudy the evolution in time of the probability density function of passive tracers in such a flow. Wefind that at initial times the tracer distribution closely follows the nonlinear background flow andthat nontrivial Stokes drift ensues as a result. Over finite times, we measure chaotic mixing basedon the stochastic mean flow and identify nontrivial mixing structures of passive tracers, as comparedto their absence in the deterministic flow. At later times, the probability density field spreads out tosample larger regions and the mean Stokes drift approaches an asymptotic value, indicatingsuppression of Lagrangian mixing at long time scales. © 2010 American Institute of Physics.�doi:10.1063/1.3518137�

I. INTRODUCTION

The study of mixing structure and transport phenomenain a nonlinear, chaotic fluid flow can be traced back to thework of Taylor �1934�,1 where he studied the emulsion of afluid droplet inside another fluid environment subject tobackground strain and shear. Later, Welander �1955�2

stressed that mixing processes should be understood by thestretching and folding of material elements. Ottino �1989�3

provides a wonderful summary of topics in chaotic mixing.In recent studies, focus has been placed on characterizationof flow structures aiming at identifying the topology of anonlinear, chaotic flow. In chronic orders, Eulerian coherentstructures �Okubo �1970�, Chong et al. �1990�, Weiss �1991�,Jeong and Hussain �1995��4–7 were first developed to extractflow regions that enhance mixing �regions of high strain� andthose that inhibit mixing �regions of high vorticity�. Haller�2000, 2001, 2005�8–10 and Haller and coauthors �2000,2003�11,12 developed the theory of Lagrangian coherentstructures �LCS�, capable of the objective extraction of mix-ing structures �hyperbolic, parabolic, and elliptic regions� intime aperiodic, chaotic flows. Most importantly, LCS identi-fies the distinguished material lines/surfaces that are mostconducive to enhancement or inhibition of the mixing oftracers, and thus they are exactly the topological structuresthat Welander �1955�2 emphasized.

In most of the studies on LCS to date, the focus has beenon their extraction in deterministic flows. There the chaotictrajectories follow a determined fashion based on the back-ground flow and hence passive tracers will follow the dy-namics organized by the LCS, provided that the evolution ofthe flow structures is much slower than the evolution oftracer trajectories �Haller �2001��.9 Geophysical flow is oneexample where the LCS theory is valid. As such, Sapsis and

Haller �2009�13 and Tang et al. �2010�14 discussed the appli-cation of LCS in identification of mixing structures in hurri-cane Isabelle and a subtropical jet stream near Hawaii. How-ever, with geophysical applications in mind �as with manyother applications�, the velocity information is not well re-solved due to constraints on the data acquisition �measure-ments or simulations� and thus there are inherent subgridscale uncertainties that affect the dynamics of a tracer. Assuch, Olcay et al. �2010�15 studied the influence of randomnoise on LCS for a vortex ring field through a Lagrangianparticle tracking method, assuming Gaussian white noise.

In this paper, we seek to obtain the stochastic mixingstructure of a nonlinear background flow subject to aniso-tropic white noise. In addition to computing the Lagrangiantrajectories by the ensemble mean of a cluster of particles,we track the evolution of the probability density fieldthrough the Fokker–Planck �FP� equations �Sobczyk�1991��.16 With its solution we will be able to construct meantrajectories and higher order moments, hence better charac-terize the influence of stochasticity on Lagrangian mixing.

We focus on studying the advection of passive tracers inan inertia-gravity wave �IGW� field. There are several moti-vations of the flow field under consideration. On the physicalend, IGWs are ubiquitous in the environment �Garzoli andKatz �1981�, Eckermann and Vincent �1993�, Plougonvenet al. �2003�, Lane et al. �2004��.17–20 Their generation, ad-vection, nonlinear interaction, and dissipation are associatedwith large energy and momentum transfer and these dynami-cal processes play important roles in the global energy bud-get of the atmosphere and ocean circulations. At smallerscales, the breaking of IGW in the upper-troposphere andlower stratosphere �UTLS� is known to form clear-air turbu-lence, a primary source of aviation hazard. The study of co-herent motion of tracer dynamics in such fields can thus out-

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line regions of instability, likely candidates for aviationhazards �Tang et al. �2010��.14 The study of Lagrangiantransport processes in this region also leads to better under-standings of the structures of isentropic and vertical mixing�Legras and d’Ovidio �2007��.21 Indeed, dynamics, chemis-try, microphysics, and radiation are fundamentally intercon-nected in the UTLS region �Mahalov and Moustaoui �2009,2010��.22,23 As such, characterization of scalar mixing andtransport via random processes in this region are very impor-tant for development of improved models. On the math-ematical end, IGW is a nice prototypical model for the studyof random noise on Lagrangian dynamics, as the mean mo-tion over a wave cycle, the Stokes’ drift �Craik �2005�,Stokes �1847��,24,25 is exactly zero when no noise is present.Therefore, any net effect as derived from the study of the FPequations would directly imply the impact of stochasticity.Through studies of the FP equations we will be able to char-acterize the mean and higher order moments of this randomprocess.

The stochastic effects on Stokes drift of Lagrangiantracer particles have been first studied by Jansons and Lythe�1998�.26 They examined the dynamics of Lagrangian tracerparticles subject to Gaussian white noise in 1D multichro-matic wave flows and find analytical expressions for theStokes drift subject to the random noises. Following theirstudies, Jansons �2007�27 studied the stochastic Stokes driftfor inertial particles. For geophysical flows, wave-generatedtransport associated with stochasticity was studied in Re-strepo and Leaf �2002�,28 where the drift velocities for pro-gressive and standing 2D waves were obtained by numericalsimulations. Stochastic Lagrangian drifts have also been in-corporated in wave driven circulation models in Restrepo�2007�29 to parametrize wave breaking effects. In this study,our focus is both on obtaining the stochastic Stokes drift ona monochromatic IGW so as to obtain the Lagrangian mixingtopology associated with stochasticity and quantifying thestatistics of the various moments for this random process.

Being able to characterize the non-Gaussianity �intermit-tency� in nonlinear processes is important in many disci-plines, including studies on Lagrangian tracer dynamics. Itprovides quantitative information of how a process is differ-ent from zero-mean, symmetric, Gaussian processes. Thisknowledge is important in improving both the modeling oftracer mixing and the detection of tracer transport in nonlin-ear flow fields. Such characterizations are based on the studyof probability density functions. A general observation fromexperimental and numerical data is the broadening of the tailof a Gaussian. Numerous studies have been dedicated intothe characterization of this broadening effect. To name a fewof these studies among a vast literature, in Bronski andMcLaughlin �2000�,30 asymptotic behaviors of large mo-ments were obtained analytically for a random linear shearmodel. Bourlioux and Majda �2002�31 evaluated intermit-tency of passive scalars associated with a mean gradient andfind four asymptotic regimes of intermittency based onchoices of the Peclét number and the flow forcing period.Kramer et al. �2003�32 carried out a comparative study onclosure approximations for passive scalar intermittency in aclass of shear models. Sukhatme �2004�33 analyzed the re-

gime that probability density functions exhibit strange eigen-modes �attain self-similarity in finite-time�. Tartakovsky etal. �2009�34 analyzed intermittency in reacting flows arisingfrom uncertainties in reaction rates. For atmospheric flows,in the stratosphere, the probability density for concentrationand tracer gradients have been studied in Hu and Pierrehu-mbert �2001, 2002�.35,36

In this paper, to study the evolution and the mixing pat-terns of a passive tracer, we first solve the FP equation overone wave cycle to obtain the deviation from the deterministicsolution. Mean drifts are obtained from the expectation ofdisplacements. Indeed, we start the simulation with differentinitial conditions to measure chaotic mixing of different trac-ers as induced by mean drifts. LCS �absent in the determin-istic flow� is obtained through the computation of finite-timeLyapunov exponents �FTLE� �Haller �2001��9 on the stochas-tic mean trajectories. The probability density fields are char-acterized by discrete moments, including variance, skewness,and kurtosis. In addition to the dynamics over a single wavecycle, we run a suite of simulations with different variancesto longer times when the expectations reach some asymptoticvalue. This allows us to study the long-time behavior of atracer and the dependence of statistics on different variances.Our solutions from the FP equation are tested against La-grangian particle tracing methods to ensure their fidelity. Infact, it is possible to obtain the analytical expressions fordifferent moments because we have a closed system due tothe form of the IGW. We only show the analyses for the firstand second moments in this paper and compute higher ordermoments numerically using the probability density calcu-lated from the FP equation.

The rest of the paper is organized as follows. In Sec. IIwe introduce the nondimensional FP equation proper to ourproblem. In Sec. III we discuss some analytic results for themean trajectories. In Sec. IV we discuss numerical resultsfrom computation of the FP equations and the Lagrangianparticle tracking methods. In Sec. V we draw conclusionsand discuss future directions. The detailed derivations ofanalytical expressions are given in the Appendix.

II. MATHEMATICAL FORMULATION

The linear solution of an IGW is given by a polarizedvelocity field �Gill �1982��37

u = �u,v,w� = �u0 cos �,u0 f�

sin �,− u0k

mcos �� , �1�

where u0 is a velocity scale, k and m are the horizontal andvertical wavenumbers, f is the Coriolis frequency, � is thewave frequency, and �=kx+mz−�t is the wave phase. Re-alistic values of these parameters for atmospheric flows aregiven later in the text. For a deterministic flow, the Lagrang-ian particle trajectories can be integrated from Eq. �1��Stokes �1847�, Lighthill �1979��25,38 and it can be shownthat the Lagrangian trajectory is exactly the same as the timeintegral of the Eulerian velocity at all times, due to the exactcancellation of kx and mz in �. Therefore, no Lagrangianmixing of tracers occurs after each wave period for the de-terministic flow.

126601-2 Tang, Taylor, and Mahalov Phys. Fluids 22, 126601 �2010�


In this paper, we consider perturbations to Eq. �1� byanisotropic, homogeneous Gaussian white noise with con-stant variances �h

2 ,�v2 along the horizontal and vertical axes.

These are related to eddy diffusivities as �h2=2�h, �v

2 =2�v.The choice of anisotropy is motivated by turbulent motion indensity stratified flows driven by shear. Specifically, in a den-sity stratified environment, vertical fluctuations are sup-pressed due to the energy required to overcome stable strati-fication. However, it is pointed out in Sarkar �2003�39 that,although suppressed by stratification, vertical motion is stillcoupled with the horizontal components of the fluctuations,and thus we still consider the full three-dimensional flowfield, with the scales of fluctuations imposed by the appro-priate choices of variances. Of course, this particular paperdoes not fully capture the nonstationary, nonhomogeneousnature of small-scale turbulent motions. Our goal is to char-acterize the heterogeneity of tracer mixing that is caused byrandomness in the nonlinear velocity data. As will be dem-onstrated in the following sections, even the simplest sto-chastic process such as Gaussian white noise can induce suchheterogeneity. In addition, with this simple noise structurewe can derive analytic solutions for the mean trajectories,which can be used to confirm the accuracy of the numericalsimulations.

Suppose that the stochastic trajectory of a tracerxt= �xt ,yt ,zt� satisfies the following system of stochastic dif-ferential equations:

dxt = udt + �hdWt�1�, dyt = vdt + �hdWt

�2�,

�2�dzt = wdt + �vdWt

�3�,

where �u ,v ,w� are given in Eq. �1� and W= �W�1� ,W�2� ,W�3�� is a standard vector Wiener process Wtwhose components are independent from each other. Thejoint probability density P of the stochastic velocity field isa solution to the FP equation, given as �Sobcyzk �1991��16

Pt + u · �P =12�h

2�h2P + 12�v

2Pzz. �3�

Using the characteristic horizontal length scale Lh and thetime scale of a wave period T=2� /�, the nondimensionalFP equation is

P� + U�cos �PX + sin �PY − cos �PZ�

= 12Dh�H2 P + 12DvPZZ, �4�

where �= t /T is the nondimensional time, U=u0T /Lh is thenondimensional velocity scale, �=2��X+Z−�� is the non-dimensional wave phase, �X ,Y ,Z�= �x ,y ,Rz� /Lh is the non-dimensional coordinates, R=m /k is the aspect ratio, and�Dh ,Dv�= ��h

2 ,R2�v2�T /Lh

2 is the nondimensional variances in

the horizontal and vertical directions, respectively. With thesimple noise structure, Eq. �4� is an advection-diffusionequation for the IGW.

The nonlinear velocity field makes the probability den-sity P in Eq. �4� analytically intractable for positive values ofthe variances. However, some averaged quantities, such asthe mean and variances, can be obtained through the hierar-chy of moments outlined in Young et al. �1982�,40 with thedefinition of an advected coordinate rotating with the flow. Intheir paper, the advection-diffusion of a polarized velocityfield rotating in the horizontal plane was considered and theproblem was solved in the context of extracting effectivediffusivity through shear dispersion. The difference amongwave phases was deliberately filtered out through the choiceof a line source. One of our aims in this study is indeedexamining the nontrivial mixing pattern arising from differ-ent phases of the IGW. As such, our initial conditions arehighly localized and the polarized flow is tilted from Younget al. �1982�.40 In consideration of higher order moments, wenote that the method of moments becomes progressivelymore difficult as the order of the moment increases. Hence,Eq. �4� is also solved numerically to obtain a more completepicture of the statistics. Mean trajectories obtained fromP are compared to analytic solutions obtained from an ex-plicit solution to Eq. �2� to ensure accuracy of the numericalsolutions.

As mentioned above, since the nontrivial stirring patternof an IGW coupled with stochasticity may give rise to anontrivial mixing pattern of tracers, we examine this mixingthrough the mean trajectories. Olcay et al. �2010�15 studiedthe influence of random noise on Lagrangian mixing, wherestochasticity is introduced by adding Gaussian white noise tothe background velocity field for a set of initial conditionsstarting at the same location and time �Lagrangian particletracking method�. The ensemble of trajectories is consideredin the computation of FTLE. To the best of our knowledge,this paper is the first to evaluate LCS due to random noisefor 3D flows subject to anisotropic perturbation and withconsideration of higher moments. Here, the FTLE are evalu-ated from the mean trajectories after one wave cycle. Spe-cifically, we compute

M�X0� � �E�X�T;X0��X0 T �E�X�T;X0��X0 ,�5�

FTLE�X0� =1

2Tlog max�M� ,

where M is the Cauchy–Green strain tensor, X0 is the initiallocation at �=0, E�X�=��PXdXdYdZ is the expectation ofthe initial condition after one wave period, �¯ T denotes thetranspose of the deformation matrix, and max evaluatesthe largest eigenvalue of M �Haller �2001��.9 The FTLE

126601-3 Lagrangian dynamics in stochastic inertia-gravity waves Phys. Fluids 22, 126601 �2010�


field highlights regions in physical space where intenseLagrangian mixing occurs during the finite-time period ofevaluation.

III. ANALYTIC SOLUTIONS

We present in this section the analytic solutions to thefirst and second order moments of the governing system. Thedetails of the derivation can be found in the Appendix. Re-writing Eq. �2� in nondimensional coordinates, we have

Ẋ� = U cos � + Dh1/2Ẇ�

�1�, Ẏ� = U sin � + Dh1/2Ẇ�

�2�,

�6�Ż� = − U cos � + Dv

1/2Ẇ��3�,

where q̇�dq /d� for some quantity q, U is the nondimen-sional velocity scale, �=2��X+Z−�� is the wave phase, andX ,Y ,Z are the nondimensional coordinates.

Considering a tracer with some initial distribution attime �=0, we find that the mean position of a tracer at time� is

X̄ � E�Xs�� = EX +UE22�

�D − exp�− 2�2D��D cos 2�� − sin 2��1 + �2D2

+UE12�

1 − exp�− 2�2D��cos 2�� + �D sin 2��1 + �2D2

,

Ȳ � E�Ys�� = EY −UE22�

1 − exp�− 2�2D��cos 2�� + �D sin 2��1 + �2D2

+UE12�

�D − exp�− 2�2D��D cos 2�� − sin 2��1 + �2D2

, �7�

Z̄ � E�Zs�� = EZ −UE22�

�D − exp�− 2�2D��D cos 2�� − sin 2��1 + �2D2

−UE12�

1 − exp�− 2�2D��cos 2�� + �D sin 2��1 + �2D2

,

where D=Dh+Dv, E1=E�sin �0, and E2=E�cos �0 denotethe expectations with respect to the possibly random initialphase, while EX=E�Xs�0�, EY =E�Ys�0�, and EZ=E�Zs�0�denote the means of the initial coordinates of the particle.

As an illustration, for the simple case where the initialevaluation of tracer position X0�0 with probability one, wehave EX=EY =EZ=E1=0 , E2=1. Thus the expression of themean trajectory Xs�� is

X̄ =U

2�

�D − exp�− 2�2D��D cos 2�� − sin 2��1 + �2D2

,

Ȳ = −U

2�

1 − exp�− 2�2D��cos 2�� + �D sin 2��1 + �2D2

,

�8�

Z̄ = −U

2�

�D − exp�− 2�2D��D cos 2�� − sin 2��1 + �2D2

.

For comparison, we observe that the particle trajectory start-ing from X0=0 in the nondimensional deterministic velocityfield is

X�� = �− U sin �/2�,U cos �/2�

− U/2�,U sin �/2�� . �9�

For very small variances Dh ,Dv1 and over small

times, the mean trajectory remains close to the deterministictrajectory Eq. �9�. Furthermore, as �→�, we observe that themean trajectory approaches �0,−U /2� ,0�, the center of theellipse where the deterministic trajectory resides. This is be-cause the phase ��, defined as

�� = 2��X0 + Z0 − �� + 2�Dh1/2W�

�1� + 2�Dv1/2W�

�3�

= �0 − 2�� + 2�D1/2W�, �10�

where Wt= �Dh /D�1/2Wt�1�+ �Dv /D�1/2Wt

�3�, evolves as aBrownian motion with a constant negative drift, which is theadvected coordinate. As time progresses, the distribution ofthe phase �� mod 2�� tends to the uniform distribution on�0,2��, so that the mean position of an ensemble of inde-pendent particles governed by the stochastic vector field isasymptotic to the time average of the position of a determin-istic particle over the course of a single cycle. On theother hand, in the limit of large variance Dh ,Dv�1, themean trajectories remain near the origin, indicating that thestochastic process behaves like a Brownian motion and isonly weakly influenced by the nonlinear background flow.For intermediate values of Dh ,Dv, the mean tracer tra-jectory will approach �UD /2�1+�2D2� ,−U /2��1+�2D2� ,−UD /2�1+�2D2��.

Analytical expressions for the second order moments are



Var�X� = EX2 − X̄2 + Dh� + UR2EXi�

�e� − 1� −U�

+

U

2�e� − 1� +

UE2i�

� −

� e�� − 1

�

−e� − 1

�

+ 4�iDhEi�� e�

−e�

2+

1

2� ,

Var�Y� = EY2 − Ȳ2 + Dh� + UJ2EYi�

�e� − 1� −Ui�

+

Ui

2�e� − 1� −

UiE2i�

� −

� e�� − 1

�

−e� − 1

� ,

Var�Z� = EZ2 − Z̄2 + Dv� + UR− 2EZi� �e� − 1� − U� + U2 �e� − 1� + UE2i�� − � e�� − 1� − e� − 1 �− 4�iDvEi�� e� − e

�

2+

1

2� ,

�11�

Cov�X,Y� = EXY − X̄Ȳ + UREYi�

�e� − 1� + UJEXi�

�e� − 1�

+ UJ UE2i�

� − � e�� − 1

�−

e� − 1

� + 2�iDhEi�� e�

−

e�

2+

1

2� ,

Cov�X,Z� = EXZ − X̄Z̄ + UREZi� − EXi�

�e� − 1�

+ URU�

−

U

2�e� − 1� −

UE2i�

� −

� e�� − 1

�

−e� − 1

� + �2�i�Dv − Dh�Ei�� e� − e�2 + 12� ,

Cov�Y,Z� = EYZ − ȲZ̄ + UJEZi�

�e� − 1� − UREYi�

�e� − 1�

+ UJ− UE2i�

� − � e�� − 1

�−

e� − 1

� + 2�iDvEi�� e� − e�2 + 12� ,

where =−�2�2D+2�i�, �=−�8�2D+4�i�, Ei�=E1+ iE2=E�sin �0+ iE�cos �0, E2i�=E�sin 2�0+ iE�cos 2�0,and EX2, EY2, EZ2, EXY, EXZ, EYZ, EXi�, EYi�, and EZi� are theexpectations of X2, Y2, Z2, XY, XZ, YZ, Xei�, Yei�, and Zei�

at time �=0, respectively.As seen, over long times, the variances scale linearly

with � and are augmented by U2D� /2��2D2+1� due to thenonlinear background flow. Considering eddy diffusivities ingeophysical flows, where vertical diffusion is strongly lim-ited by density stratification �DvDh�, the above expressionimplies that Var�Z� remains finite due to horizontal diffusion.In this limit, the covariances Cov�X ,Y� and Cov�Y ,Z� alsostay finite while Cov�X ,Z� decreases linearly with � due tothe symmetry in X and Z.

IV. NUMERICAL RESULTS

The probability density P naturally satisfies vanishingboundary conditions at infinity. To ensure that the problem isnumerically tractable with high precision, we solve Eq. �4� ina finite domain subject to periodic boundary conditions andrequire P to be negligibly small near the computational

boundaries compared to its maximum value inside the do-main for all time �the ratio between them is maintained at10−8 for all time�. The numerical solver we use is describedin detail in Bewley �2011�.41 Since periodic boundary condi-tions are applied in all directions, derivatives are treated witha pseudospectral method. The low storage third-orderRunge–Kutta–Wray method was used for time stepping anddiffusive terms are treated implicitly with the Crank–Nicolson method. In order to prevent spurious aliasing due tononlinear interactions between wavenumbers, the largest 1/3of the horizontal wavenumbers are truncated using the 2/3dealiasing rule �Orszag �1971��.42 The initial probability dis-tribution was taken to be Gaussian with density P satisfying��PdXdYdZ=1 and the nondimensional variances of Pwere �X

2 =�Y2 =�Z

2 =�02=0.005. The differential equation is

then integrated in time over one wave period T and longer toobtain the statistics of the randomness of the wave field. Weuse variance, skewness, and excess kurtosis to measure thestatistics of the probability density. For Brownian motion, thevariance grows linearly with �. Skewness measures theasymmetry of the probability density and is zero for a per-



fectly symmetric field. Excess kurtosis measures the flatnessof the probability density. It is zero for the Gaussian and �3for a uniform probability density. Any deviation from thesestandard values indicates deviation from a Gaussian process.In addition, we compute the mean trajectories initiated fromdifferent initial conditions to characterize the Lagrangianmixing structure. Note that due to the periodic nature of thewave field, the transition density P can be obtained for anyinitial location by simply varying the initial phase �0. This isused when constructing the Lyapunov exponents of the meantrajectories.

As discussed in Sec. I, we consider an IGW generated bythe tropospheric midlatitude jet stream �Plougonven et al.�2003��.19 In this setting, a typical velocity scale is u0�7 m /s, wave period T�12.2 hr, horizontal length scaleLh�220 km, and aspect ratio R�100. Stochasticity is intro-duced through eddy diffusivities estimated for the UTLS.The vertical eddy diffusivity near the lower stratosphere, asdiscussed in Wilson �2004�,43 varies from O�0.01��O�1� m2 /s. For the primary case I, we use �v=0.1 m2 /s,in line with observations outside the polar vortex �Legras etal. �2003��.44 The horizontal eddy diffusivity has strongervariability and we use the value �h=1000 m

2 /s. In additionto case I, we examine the effects of different diffusivities onthe mean trajectories by simulating the probability densityfor three progressively larger diffusivities, while holding theratio between the horizontal and vertical diffusivities con-stant. The choices of these additional cases are still in physi-cally realizable ranges. These additional simulations revealhow different diffusivities can affect the statistics of tracerdynamics in IGW.

A. Statistics after one wave cycle

We assume that the initial Gaussian probability densityis centered at the middle of the computational box and solveEq. �4� over one wave period. The temporal evolution of thestatistics of an initial condition at wave phase �0=0 isshown in Fig. 1. Unless indicated otherwise, solid curvesdenote measures in X, dashed curves denote measures in Y,and dashed-dotted curves denote measures in Z. We note thatfor some initial probability distributions, the nonlinear back-ground flow is capable of advecting the probability density

around and driving it away from a Gaussian distribution dur-ing a wave cycle, even without stochasticity. The analyticalsolution for the advected probability density, started from anisotropic Gaussian distribution, is

P�X,�� =1

�2��3/2�03exp�− �X̃�22�02� , �12�

where

X̃ = X + �U sin �/2�,− U cos �/2� − U/2�,

− U sin �/2�� 13�

is the advected coordinate that corresponds to the initial lo-cation X0 of a tracer which is at location X at time � �Younget al.�1982��40 and �X̃�2 denotes the Euclidean norm of theadvected coordinate. This expression is valid because theinitial standard deviation is isotropic. For comparison, weuse thick curves to indicate the simulation results from caseI and thin curves to indicate the deterministic case. As seenin Fig. 1�a�, the time dependent standard deviation � for Xand Z is at its maximum at half period T /2, whereas the timedependent standard deviation for Y has maxima around T /4and 3T /4. This is not surprising, since for two trajectoriesinitiated near �0=0, cosine function creates the largeststretching half way through a period, whereas the sine func-tion reaches extremes at T /4 and 3T /4. For the stochasticcase, the observations that the second peak in �Y is largerthan the first peak and that the standard deviations return tovalues larger than their initial conditions are reassuring sincestochastic processes should work to increase variance. Thistrend is confirmed from comparison between the thick andthin curves.

The temporal evolution of the skewness S and the excesskurtosis K also show that there is distortion away from aGaussian distribution. To be exact, the case with zero sto-chasticity indicates a distortion of the Gaussian structure dueto the nonlinear background flow and this non-Gaussianstructure evolves over a wave cycle to return to the initialGaussian structure instantaneously. Stochasticity works to re-duce the non-Gaussian behaviors of S and K created by thenonlinear background flow, even though the statistics showthat the distribution is not Gaussian at the moment of the

0 0.5 10.05

0.1

0.15

0.2

0.25

0.3

�

�

0 0.5 1�1

�0.5

0

0.5

1

1.5

�

S

0 0.5 1�1

0

1

2

3

�

K

a) b) c)

FIG. 1. �a� Temporal evolution of variance � of the probability density P for initial phase �0=0. The thick version of solid, dashed, and dashed-dotted curvesdenote the standard deviations in the X, Y, and Z directions, respectively. The thin version of these curves shows respective variances computed from a casestarted from the same initial conditions but with no diffusion. �b� Temporal evolution of the skewness S. �c� Temporal evolution of excess kurtosis K. The linestyles of �b� and �c� are the same as �a�.



completion of a wave cycle. As we show later, stochasticityprevails in the long run in the determination of the hierarchyof moments, and all statistics indicate a Gaussian processwhose skewness may be affected by the initial conditions.

Next we consider Lagrangian stirring induced by nearbytracers. Because of the spatial asymmetry, we do not expectthe mean trajectories to return to their initial locations. Assuch, nontrivial Lagrangian mixing will occur as comparedto the deterministic case with no Lagrangian mixing. Wecompute the mean trajectories E�X, E�Y, and E�Z startingfrom different initial conditions and integrate over one wavecycle. We then use FTLE discussed in Sec. II as the measureof chaotic mixing to characterize the stochastic stirring offluid particles. From Eq. �7� we expect the end locations ofmean trajectories for different initial phases after one wavecycle to behave as trigonometric functions, since the evolu-tion of the mean trajectory at �=1 is only a function of �0,implicitly embedded in EX, EY, EZ, E1, and E2. As shown inFig. 2�a�, for case I, the mean trajectory can be described as

E�X = − A sin �0, E�Y = − A cos �0,�14�

E�Z = A sin �0,

where A�0.013 and �0 is the initial phase a tracer assumesat time 0. This significantly simplifies the computation of theFTLE, since the largest eigenvalue of M takes the form

= 1 + A2�1 + cos �02�

+ �A4�1 + cos �02�2 + 2A2�1 + cos �02� . �15�Starting from a uniform set of grid points, the initial condi-tions are deformed by the flow and we can locate the mate-rial surfaces that attract or repel nearby trajectories. We showthe deformation of these initial conditions after one wavecycle in Fig. 2�b�. The repellers are highlighted by the threesolid lines. Note that we have exaggerated the values ofmean trajectories by a factor of 5 to make the repellers vis-ible. Since the FTLE field is independent of Y, we only showthe X−Z section of this scalar field in Fig. 2�c�. EnhancedLagrangian mixing is found along the half-integer phase line�dark shaded regions, color contour online�, which deviatesfrom the trivial mixing case of a deterministic flow.

It is worth mentioning here that even though we haveperiodicity in our system and thus can extract the infinitetime Lyapunov exponent, we still only focus on its finite-time counterpart. The reason is that for the spatial structureof Lagrangian mixing, it is the geometry, rather than theexact value of Lyapunov exponent, that plays importantroles. Because of the periodic behavior of the mean trajecto-ries, the geometry we obtain from finite-time is the same asthat computed from infinite time. However, for physicallyrelevant tracers, such as ozone, their chemical properties willeventually become important over long time scales, hencethe infinite time Lyapunov exponent will not correctly char-acterize behaviors over infinite times.

B. Different variances and long-term behavior

In order to evaluate the effects of different variances onthe mean trajectories, we ran four cases initialized at phase�0=0 with the variances listed in Table I. The case with zerovariance is included for reference. The results of these simu-lations are summarized in Fig. 3. The thick solid circle inFig. 3 shows the mean trajectory associated with the deter-ministic flow with no stochasticity, as described in Eq. �12�.The mean trajectories over one wave cycle for different vari-ances are shown as dashed-dotted curves inside the circle,with their end positions marked by the dots. As expected, themean trajectories move away from the big circle as varianceis increased. Indeed, because of the different diffusion timescales associated with the different variances, comparisonbetween the trajectories at one fixed time period is lessmeaningful than the asymptotics. �This, however, does notinvalidate the evaluation of FTLE at one wave period, as

FIG. 2. �Color online� �a� Mean trajectory E�x ,E�y and E�z for different initial phases. Line styles are the same as in Fig. 1. �b� End positions of uniformlyspaced initial conditions after one wave cycle. Solid lines indicate repelling material surfaces. �c� FTLE computed based on Eq. �15�.

TABLE I. Different variances used in simulations.

Case�h

2

�m2 /s��v

2

�m2 /s�

0 0 0

I 2000 0.2

II 6324 0.6324

III 20 000 2

IV 63 240 6.324



Lagrangian mixing is already taking place within finite time.�As such, we continue the simulation until the mean trajecto-ries settle to �approximately� a single point and plot them asthe crosses inside the circle. For clarity and reference, wecontinue to plot the mean trajectory for case IV with thelargest variance. The asymptotic position of this trajectory isthe furthest from the center of the circle among the fourcases considered. As we move toward smaller variance, themean trajectory settles toward the center of the circle. Ofcourse, when the variance is very large, the asymptotic pointwill coincide with the starting point, as it diffuses too rapidlyand does not feel the nonlinear background flow field. Forcomparison, we also plot in Fig. 3�a� the asymptotic posi-tions computed analytically from Eq. �7� as the dashed-dotted curve, assuming the same initial probability as in thesimulation. Here, EX=EY =EZ=E1=0 , E2=exp�−2�2��X

2

+�Z2��=exp�−0.02�2�. Thus, mean trajectories in the simula-

tion starting from E�X�0�=0 will asymptote to positionUE2��D ,−1 ,−�D� /2��3D2+��=U exp�−0.02�2��D , −1 ,−�D� /2��3D2+��. It is apparent that the crosses from thesimulation fall exactly onto the dashed-dotted curve, as ex-pected.

We are interested in learning other statistics of the tracerdynamics over long-time. In Sec. III we obtain analyticalexpressions for second order moments. These analyticalexpressions are compared against the numerical simulationsin Figs. 3�b� and 3�c�. Using the initial Gaussian profile,we find that Ei�=exp�−0.02�2�, E2i�=exp�−0.08�2�,EXi�=0.01�i exp�−0.02�2�, EYi�=0, and EZi�=0.01�i exp�−0.02�2�. In Fig. 3�b�, we show the compari-son between variances Var�X� �solid�, Var�Y� �dashed�, andVar�Z� �dashed-dotted�. We also plot in Fig. 3�c� the covari-ances Cov�X ,Y� �solid�, Cov�X ,Z� �dashed�, and Cov�Y ,Z��dashed-dotted�. The thick curves are from numerical simu-lations up to �=11. Analytical expressions are computed upto �=12 and shown in thin curves. Clearly, the comparisonshows that both results are identical. Specifically, to the lead-ing order, the variances scale linearly with �. Cov�X ,Z� alsoscale linearly with � due to the X ,Z symmetry whereasCov�X ,Y� and Cov�Y ,Z� asymptote to a constant.

In Fig. 4 we show the standard deviation, skewness, andexcess kurtosis over time for case III. Figure 4�a� shows alog-log plot of the standard deviations with a straight line

�0.2 0 0.2�0.4

�0.2

0

X

Y

0 4 8 120

0.2

0.4

0.6

0.8

��2

0 4 8 12

�0.45

�0.25

�0.05

�

Covariance

a) b) c)

FIG. 3. �a� Mean trajectories for different variances assumed. The dashed curves indicate the mean trajectories over one wave cycle. The dots indicate the endpositions of these trajectories after one wave cycle. For case IV the mean trajectory is also shown to its asymptote position. The crosses indicate the asymptoticlocations of the mean trajectories for different variances. The dashed-dotted curve shows analytic results of the asymptotic positions for different variances.��b� and �c�� Comparison between analytical and numerical solutions of the variances for case III. The thick curves indicate those computed from numericalsimulations up to �=11. The thin curves are from analytical expressions �11� up to �=12. Line styles in �b� are the same as Fig. 1. Line styles in �c� are: solid�Cov�X ,Y��, dashed �Cov�X ,Z��, and dashed-dotted �Cov�Y ,Z��.

10�2 100 102

10�1

100

�

�

0 5 10�1

�0.5

0

0.5

1

�

S

0 5 10�1

�0.5

0

0.5

1

1.5

2

�

K

2 4 62

4

6

X

Y

a) b) c)

FIG. 4. �Color online� �a� Long time behavior of the standard deviation, skewness, and excess kurtosis for case III. Line styles are the same as Fig. 1. Thestraight line in �a� indicates a scaling of �1/2. The inset plot in �b� shows the color contours of the probability density field on a horizontal cut, indicating thenonvanishing skewness in the X direction.



indicating a scaling of �1/2. At the end of case III �and alsofor cases I and II, not shown�, the standard deviations ap-proach a scaling slightly smaller than �1/2. In contrast, as weshow in the next subsection, in case IV the standard devia-tions do indeed approach a scaling of �1/2 at the end. Thisobservation suggests that in the first three cases the simula-tions probably have not been run long enough for � toclosely approach the infinite time scaling. Upon examinationof other discrete moments, we find that the excess kurtosis inFig. 4�c� returns to Gaussian with K returning to 0. However,the skewness in X and Z, shown in Fig. 4�b�, asymptotes to anontrivial value that depends on the initial phase. We plot ashaded isocontour �color contour online� of the horizontalcut of P at Z=0 at the end of the 11th wave cycle, when themean trajectory asymptotes. The isocontour clearly indicatesa nontrivial skewness in the X-direction, which is not re-moved by stochasticity.

Nonzero values of the skewness can also occur if theprocess is at its intermediate time scale. To exclude this pos-sibility from the simulation results we use P to calculategeneral moments �Ferrari et al. �2001��45

��q�s� = �V

�q − E�q�sPdV �16�

for quantity q and approximate the power law ��X�s��s atlarge times. We find that �s=s /2 for s� �0,10�, consistentwith a strong self-similar, normal diffusion process. Hencethe convergence of the skewness to nonzero values indicatesthat asymmetry arises from shear dispersion in IGW.

C. Lagrangian particle tracking methods

In addition to comparison with analytic solutions, wealso check our computation of the FP equation by comparingwith results from Lagrangian particle tracking methods�Crimaldi et al. �2008��.46 For each individual case we seed105 initial conditions that take the same initial distribution as

the computation of the FP equation and obtain various statis-tics. The initial conditions are iterated forward in time sub-ject to the equation

Xi = Xi−1 + U�Xi−1��i + Z�Dk��i, �17�

where Xi is the ith iteration of a particle trajectory, U is thenondimensional velocity field, Z is a three-dimensionalGaussian process with zero-mean and unit variance, D1=D2=Dh , D3=Dv are the nondimensional variances, and��i=�i−�i−1 is the time stepping.

The ensemble of trajectories E�X=N−1� j=1N Xj, where

N=105 is the number of samples, are used to compute themean drift due to stochastic noise. The standard deviation iscomputed as ��X�=�N−1� j=1N �Xj−E�X�2 to compare withthe solution from Eq. �4�. We have computed these for allcases of diffusivities but for clarity we only show the resultsfor the case with the highest variance.

Figure 5�a� shows the comparison between mean trajec-tories computed from the FP Eq. �4� �solid curve� and fromthe Lagrangian particle tracking method Eq. �17� �dashedcurve�. As can be seen, the mean trajectories show very smalldifference until the solid curve approaches its asymptotic po-sition. At this stage the mean trajectory computed from theLagrangian particle tracking method starts to show randomnoise in the mean trajectories, indicating a decorrelation ofthe mean trajectory with the background flow. Figure 5�b�shows the comparison between the standard deviations com-puted from the different methods in log scale. The curvesdenote the computation from FP and the markers are fromLagrangian particle tracking. There is no notable differencebetween the two cases as the mean trajectory reaches itsasymptote. For reference we also plot in Fig. 5�b� the func-tion �1/2 as the straight line. The plot indicates an asymptoticscaling of the standard deviation similar to Brownian motion.

0 0.05 0.1 0.15 0.2�0.25

�0.2

�0.15

�0.1

�0.05

0

X

Y

10�2 10�1 100 101

10�1

100

�

�

a) b)

FIG. 5. �a� Comparison between the mean trajectories computed from Eq. �4� �solid curve� and from the Lagrangian particle tracking method Eq. �17� �dashedcurve� for case IV. �b� Comparison between the standard deviations for the two methods. Plotted in curves are solutions from the FP equations and in markersymbols are solutions from the Lagrangian particle tracking method. The line styles are the same as Fig. 1. For the markers, “x” denotes �X, “+” denotes �Y,and “*” denotes �Z.



V. DISCUSSIONS

In this paper we have studied the Lagrangian dynamicsof a tracer in an inertia-gravity wave field embedded withinan environment of Gaussian white noise. It can be shownthat the deterministic Lagrangian trajectories of an IGW areexactly zero over a wave cycle, and hence no Lagrangianmixing occurs in such a case. We examined the question ofwhether mean drift and nontrivial mixing structure will arisewhen stochasticity is added to the model. The study of theprobability density structure has important implications inunderstanding the Lagrangian dynamics of a passive tracer ina stochastic environment. In the case considered in this pa-per, it means that Lagrangian stirring of the idealized IGWdoes arise in a realistic environment, induced by stochasticityin a nonlinear flow.

We have formulated the problem in the context of anIGW in the upper-troposphere-lower-stratosphere, motivatedby studying tracer dynamics leading to better understandingsof the water vapor and ozone concentration, which are twoimportant green house gases in these regions. The nondimen-sional Fokker–Planck equations were solved using a pseu-dospectral direct numerical simulation �DNS� solver, wherethe probability density is confined well within the computa-tional box and the periodic boundary conditions mimic avanishing boundary condition at infinity. We find that, due tothe nonlinear background, nontrivial mean trajectories arisewhen stochasticity is considered, and the difference betweenthe stochastic mean trajectory and the deterministic trajec-tory increases with the sizes of the variances imposed on thesystem. These mean trajectories compare well with analyticsolutions. Due to this mean motion, we find that there areinitial phases where mean trajectories repel or attract nearbymean trajectories, serving as the enhancers in the stirring ofthe tracers. In the deterministic flow, however, no such struc-tures can be identified. We also studied the long-term behav-ior of trajectories and find that the trajectories asymptote topositions away from their initial conditions. Long-term be-haviors of higher order moments indicate that the process isGaussian. However, the probability density can have nonzeroskewness due to its dependence on initial phase. In addition,analytical solutions of the mean trajectories are obtained andcompared with the numerics. The good correlation betweenour analytical and numerical results gives us confidence inthe accuracy of the various statistics estimated from the nu-mericalsolutions.

We note that there are several ways to extend the currentstudy. First, we have assumed that the background flow isgiven by an idealized IGW and thus the interaction withother processes, such as the jet stream which emit theseIGW, or wave-wave interactions, should be studied. Stochas-tic Stokes drift in wave-wave interactions for 1D waves andwave-mean interactions have been discussed in Jansons andLythe �1998�26 and Restrepo �2007�,29 where the authorsonly focused on the mean drift. We will carry further analy-ses of 3D wave-mean/wave-wave interactions from both nu-merical and analytical approaches and obtain a more com-plete picture of the statistics. Such studies of tracer dynamics

in nonlinear interactions will reveal better the realistic mix-ing structure across the tropopause. Second, we have onlyconsidered a stable configuration of the IGW. When thebackground flow becomes unstable, which is usually the caseof more concern, Lagrangian mixing is enhanced by the non-linear motion even in the deterministic flow �Mahalov et al.�2008��.47 Random noise will also enter the dynamics of thewaves. We want to investigate tracer dynamics in this sce-nario of more intense mixing to better characterize the sta-tistics. Third, we have assumed that the mean flow is per-turbed by Gaussian white noise, which is not the mostphysically realistic assumption. For eddying motion at thescales considered, the processes are spatially and temporallycorrelated, hence in the future we will also consider caseswith these correlations.

Nevertheless, even with an elementary assumption ofGaussian white noise, our study demonstrates the existenceof nontrivial Lagrangian stirring of the tracers. This suggeststhat the contribution of randomness to Lagrangian dynamicsand its applications deserve more attention from scientistsinterested in the studies of transport processes in nonlinearflows.

ACKNOWLEDGMENTS

We acknowledge support from the Air Force Office ofScientific Research �Grant No. FA-9550-08-1-0055� and theNational Science Foundation �Grant No. ATM-0934592�.We also thank Bill Young for helpful insights on sheardispersion.

APPENDIX: DERIVATION FOR ANALYTIC SOLUTIONS

The mean trajectory of a particle moving under the in-fluence of the stochastic velocity field can be calculated ex-plicitly �Young et al. �1982�� Ref. 40� through the hierarchyof moments. Here we use an approach based on infinitesimalgenerators. If f�X ,Y ,Z , t� is a continuous function of its ar-guments, then the expectation u�t�=E�f�Xt ,Yt ,Zt , t�� solvesthe integral equation

u�� = u�0� + �0

�

E�Af�s��ds , �A1�

where

A�f� = f� + U cos �fX + U sin �fY − U cos �fZ

+ 12Dh�fXX + fYY� +12DvfZZ, �A2�

is the infinitesimal generator of the diffusion process thatsolves Eq. �6�. Because of the symmetry in X and Z, weobtain closed systems of equations for the moments of thisprocess due to cancellations of higher order terms in thegenerator.

To calculate the first order moments, let fX=X, fY =Y,and fZ=Z and observe that

A�fX� = U cos �, A�fY� = U sin �, A�fZ� = U cos � .

�A3�

Thus, taking the expectations, we have



E�X� = E�X0 + U�0

�

E�cos �sds ,

E�Y� = E�Y0 + U�0

�

E�sin �sds , �A4�

E�Z� = E�Z0 − U�0

�

E�cos �sds .

To calculate the expectations appearing inside the integrals,we observe that conditional on the initial values X0 and Z0,and �s is normally distributed with mean 2��X0+Z0−s� andvariance 4�2Ds �cf. Eq. �10��. Thus, if E1=E�sin �0 andE2=E�cos �0 denote the expectations with respect to thepossibly random initial phase, then a little calculus showsthat

E�ei�s = �E2 + iE1�e−�2�2D+2�i�� = Ei�e

−�2�2D+2�i��. �A5�

Ei�=E2+ iE1=E�ei��0� is introduced here for use in the deri-vation of higher order moments. These expressions can thenbe substituted back into Eq. �A4� to solve for the mean po-sition of the tracer at time �, given by Eq. �7�. Note thesimilarities and differences between the current approach andmethod of moments outlined in Young et al. �1982�.40 Intheir paper, horizontal averages of tracer concentration areobtained and used to solve for higher moments. Here theexpectations E�sin �s and E�cos �s are elementary quanti-ties in constructing the first moment.

In order to compute the second order moments, we needto evaluate the generator on quadratic functions of X ,Y ,Z.Writing fXX=X

2, fXY =XY, etc., we obtain

A�fXX� = 2UX cos � + Dh,

A�fYY� = 2UY sin � + Dh,

A�fZZ� = − 2UZ cos � + Dv,�A6�

A�fXY� = UY cos � + UX sin �,

A�fXZ� = U�Z − X�cos �,

A�fYZ� = UZ sin � − UY cos � .

Thus, we also need to calculate the expectations of X cos �,X sin �, Y cos �, Y sin �, Z cos �, and Z sin � to proceed.Writing these terms in the form of complex exponentials,eX�s�=Xei�s, etc., we have

A�eX�s�� = − �2�2D + 2�i�Xei�s +U

2�1 + e2i�s�

+ 2�iDhei�s,

A�eY�s�� = − �2�2D + 2�i�Yei�s +Ui

2�1 − e2i�s� , �A7�

A�eZ�s�� = − �2�2D + 2�i�Zei�s −U

2�1 + e2i�s�

+ 2�iDvei�s.

Taking the expectation of Eq. �A1� and differentiatingwith respect to time, the solution to Eq. �A7� can be found bysolving three systems of linear first order ODEs. These threesystems only differ in their forcing terms. Following Eq.�A5�, we first obtain the expectations

E�e2i�s = E2i�e−�8�2D+4�i�s, �A8�

where E2i�=E�e2i��0�.Using g to represent the unknowns and b to represent the

forcings of the individual systems, we have

ġ = g + b , �A9�

where =−�2�2D+2�i�. Individual choices of �g ,b� are oneof the following three groups: �E�Xei�s ,2�iDhE�ei�s+U�1+E�2i�s� /2�, �E�Yei�s ,Ui�1−E�e2i�s� /2�, and�E�Zei�s ,2�iDvE�ei�s−U�1+E�2i�s� /2�.

The solution to Eq. �A9� is

g�� = e��E�g�0� + �0

�

e−sb�s�ds� . �A10�Using these results we find the analytic expressions for thesecond order moments shown in Sec. III. For example, usingEX2 and EXi� to denote the initial expectations E�X2�0�,E�Xei��0�,

Var�X� = E�X2 − X̄2

= EX2 − X̄2 + Dh� + 2U�

0

�

E�X cos �d��

= EX2 − X̄2 + Dh� + 2U�

0

�

R�E�Xei��d��,

=EX2 − X̄2 + Dh� + 2U�

0

�

R

�e��EXi� + �0

��e−sb�s�ds�d��, �A11�

where X̄ is the mean trajectory defined in Eq. �7� and R�¯ denotes the real part. Using the values from Eqs. �A5� and�A8� we obtain the following expression for the variance:

Var�X� = EX2 − X̄2 + Dh� + UR2EXi�

�e� − 1� −U�

+U

2�e� − 1� +

UE2i�

� −

� e�� − 1

�

−e� − 1

�

+ 4�iDhEi�� e�

−e�

2+

1

2� , �A12�

where is as defined earlier and �=−�8�2D+4�i�.As seen, the derivations for analytic expressions become

increasingly complex as we move toward higher order mo-



ments. We therefore stop at the second order and resort tonumerical computations to evaluate third- and fourth-ordermoments.

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