lagrangian stochastic modeling in coastal...

JANUARY 2002 83B R I C K M A N A N D S M I T H

Lagrangian Stochastic Modeling in Coastal Oceanography

DAVID BRICKMAN AND P. C. SMITH

Department of Fisheries and Oceans, Bedford Institute of Oceanography, Dartmouth, Nora Scotia, Canada

(Manuscript received 13 November 2000, in final form 18 June 2001)

ABSTRACT

Lagrangian stochastic (LS) modeling is a common technique in atmospheric boundary layer modeling but isrelatively new in coastal oceanography. This paper presents some fundamental aspects of LS modeling as theypertain to oceanography. The theory behind LS modeling is reviewed and an introduction to the substantialatmospheric literature on the subject is provided.

One of the most important properties of an LS model is that it maintains an initially uniform distribution ofparticles uniform for all time—the well-mixed condition (WMC). Turbulent data for use in an oceanic LS model(LSM) are typically output at discrete positions by a general circulation model. Tests for the WMC are devised,and it is shown that for inhomogeneous turbulence the data output by an oceanic general circulation model issuch that the WMC cannot be demonstrated. It is hypothesized that this is due to data resolution problems. Totest this hypothesis analytical turbulence data are constructed and output at various resolutions to show that theWMC can only be demonstrated if the resolution is high enough (the required resolution depending on theinhomogeneity of the turbulence data). The output of an LSM represents one trial of possible ensemble and thispaper seeks to learn the ensemble average properties of the dispersion. This relates to the number of particlesor trials that are performed. Methods for determining the number of particles required to have statistical certaintyin one’s results are demonstrated, and two possible errors that can occur when using too few particles are shown.

1. Introduction

Lagrangian stochastic modeling of the advection anddispersion of clusters of particles has been a commontechnique in atmospheric boundary layer modeling forclose to 20 years. The situation in oceanography issomewhat different, where the use of stochastic tech-niques has predominantly been applied to the analysisand modeling of drifter dispersion in turbulence as-sumed to be homogeneous (see, e.g., Dutkiewicz et al.1993; Griffa et al. 1995; Buffoni et al. 1997; Falco etal. 2000). However, recently, the use of Lagrangian sto-chastic models (LSMs) in oceanography has increasedprimarily due to the use of individual-based models inbiophysical modeling of the early life history of fisheggs and larvae (Sclafani et al. 1993; Werner et al. 2001,1996; Hannah et al. 1998). Due to this difference inoperational prerogative, many of the basic principles ofLS modeling, especially in inhomogeneous turbulence,are not well known to oceanographers. One of the pur-poses of this paper is to provide a review of LS tech-niques as they pertain to oceanography.

U.S. GLOBEC Contribution Number 186.

Corresponding author address: Dr. David Brickmann, Departmentof Fisheries and Oceans, Bedford Institute of Oceanography, P.O.Box 1006, Dartmouth, NS B2Y 4A2, Canada.E-mail: [email protected]

In general there are three main reasons why LS mod-eling in the coastal ocean will differ from modeling inthe atmosphere. First of all, the atmosphere has a tur-bulent boundary layer at its bottom surface, while theocean has a turbulent boundary layer at its top (due towind stress) and bottom surfaces. The result is that theocean’s vertical turbulent structure is typically more in-homogeneous than the atmosphere’s, a fact that canmake modeling more difficult.

Second, LSMs of the lower atmosphere typically usea surface turbulent variable (e.g., the friction velocityu*) and analytical vertical turbulent structure functions[e.g., velocity variance s2 5 s2(z; u*)] to describe tur-bulence properties. In oceanography, the use of high-resolution circulation models with embedded turbulenceclosure schemes (e.g., Mellor and Yamada 1974) meansthat turbulent quantities for use in an LSM are outputat discrete levels in the vertical (as well as horizontalposition and time). We will see that this seemingly in-nocent difference can make accurate LS modeling vir-tually impossible.

Third, LS modeling in the atmosphere tends to be inthe category of short-range boundary layer transportstudies where high accuracy in time and space is im-portant (considered to be the main virtues of LS mod-eling), and simplifying assumptions regarding dimen-sion (2D vs 3D) and turbulent correlations are justifi-able. Ocean particle tracking tends to fall into the long-

84 VOLUME 19J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y

range transport category, with (usually) lower demandson accuracy but (possibly) requiring true 3D stochasticmodeling. This requires use of a different type of LSM,a random displacement model (RDM), the properties ofwhich will be described in section 2.

A key property of a correct LSM is that it maintainsan initially uniform concentration of particles uniformfor all time. This is called maintaining the well-mixedcondition (WMC). A major focus of this work is dem-onstrating the WMC for typical turbulent conditions inthe coastal ocean, as this indicates the principles andcaution that must be taken in such modeling. This istreated in section 3.

One run of an LSM provides one trial of a possibleensemble of trials. However, we need to know the en-semble average dispersive properties of the turbulentflow. This leads to the classic problem of stochasticmodeling: how many realizations (or particles) are need-ed to have statistical confidence in one’s results? Wewill see (section 4) that the answer to the question ofsufficient realizations or number of particles is specificto the problem under consideration.

The outline of this paper is as follows. In section 2we review the theory behind Lagrangian stochasticmodels and discuss concepts such as the Lagrangiantimescale, the well-mixed condition, the diffusion limitof an LSM, and the inhomogeneity index for 1D tur-bulence. Section 3 deals with demonstration of theWMC and possible problems in doing so. Section 4looks at the question of sufficient realizations, andshows how this can be determined, how it is specific tothe question being asked, and what errors are possiblewhen underseeding the source region. The last sectionis a summary and discussion.

2. Review of theory

The theory behind Lagrangian stochastic modeling iswell represented in the literature, and the aim of thissection is not to explain the complicated details of thederivation, but rather to present the key aspects relevantto an operational understanding of the model(s). Thereader is referred to the excellent monograph by H. C.Rodean (Rodean 1996) for a clear, expanded version ofthe derivation of LSMs. The section begins with a re-view of the derivation, then moves on to topics relatedto LSMs and their use.

a. Model derivation

The starting point for LS modeling is the 3D Langevinequation [see chapter 1 of Gardiner (1983) for an in-teresting discussion of the historical setting]

du 5 a (x, u, t)dt 1 b (x, u, t)dW (t),i i ij j (1)

with

dx 5 udt.

To simplify discussion, consider the 1D version of (1)for the vertical velocity w:

dw 5 a(z, w, t)dt 1 b(z, w, t)dW(t). (2)

In the above velocity u 5 (u, y, w), the velocity incre-ment dui 5 (du, dy, dw), x 5 (x, y, z), t is time, dt isthe time step, dx 5 dxi, and dW is the incremental‘‘Weiner process’’ with variance dt [W(t) 5 z(s) ds,t#0

dW 5 zdt, where z(t) is a ‘‘white noise’’ random forc-ing].

The first term on the right-hand side of (1) is a de-terministic term called the drift term. The second termis a stochastic or diffusion term. The goal is to derivephysically meaningful forms for the functions a and b.

Equation (2) is an example of a stochastic differentialequation, and associated with it is a probability densityfunction (pdf ) for w, P(z, w, t). The evolution of P isdescribed by a Fokker–Planck equation,

2]P(z, w, t) ](wP) ](aP) 1 ](b P)5 2 2 1 , (3)

2]t ]z ]w 2 ]w

which states that the evolution of P(z, w, t) in phasespace is due to 1) vertical advection of P, plus 2) thedivergence in phase space of aP (the drift of proba-bility), plus 3) the diffusion of P with diffusion coef-ficient b2, that is, P evolves via an advection–diffusionequation.

The Fokker–Planck equation is a differential form ofthe Chapman–Kolmogorov equation for a Markov pro-cess and required this assumption in its derivation (seeRodean (1996, ch.5) and Gardiner (1983). Simply put,a Markov process is one in which the probability of afuture state is independent of the past, depending onlyon the present plus a transition rule (pdf ) that takes usfrom the present to the future. Strictly speaking, (3)describes the evolution of the transitional (or condi-tional) pdf P(z, w, t | z0, w0, t0); the probability that aparticle initially at z0 with velocity w0 at t0 is observedin the dz, dw, neighborhood of z, w, at time t, whichby the Markov assumption is related to the uncondi-tional pdf P(z, w, t).

SOLVING FOR a AND b

The Fokker–Planck equation is the key to findingexpressions for a and b because it provides a link be-tween the probability density function of the stochasticdifferential equation and statistical properties that canbe derived from the governing Eulerian equations.

Two methods of solution are discussed in the litera-ture: the moment method [Sawford (1986); van Dop etal. (1985); also Du et al. (1994) for a recent view); anddirect solution of the Fokker–Planck equation (Thomson1987, hereafter Thomson]. They both involve the fun-damental (and intuitive) constraint that an initially well-mixed (uniform) distribution of particles remains uni-form in unsteady, inhomogeneous turbulence—theWMC. The second requirement is called Eulerian con-


sistency, which means that the statistics of the flow de-scribed by the LS model must be equivalent to the sta-tistics derived from the governing Eulerian equations.It was Thomson who showed that these two require-ments are, in fact, identical.

Thomson’s method requires two important ideas.

1) Where b has a universal form. For time incrementswithin the inertial subrange the random part of theLSM should be determined from similarity theoryconsiderations (see Wilson and Sawford 1996). Thisyields

b 5 ÏC e ,o

where e is the turbulent dissipation rate and C0 is auniversal constant O(1).

2) The WMC requires, formally, that P 5 PE; that is,the pdf of the LSM is the same as the pdf for thegoverning Eulerian field (Thomson 1987). As well,the only consistent pdf is one that has vanishing oddmoments (Sawford 1986); that is, PE(z, w, t) isGaussian,

1 2G 21/2(w /s )wP 5 P (z, w, t) 5 e ,E E2Ï2psw

where 5 (z, t) is the variance of w; the pdf is2 2s sw w

everywhere Gaussian, but its variance is a functionof z and possibly t.

b. The solution of the 1D equation is unique

Using the above, Thomson found a unique solutionto the 1D problem. Writing it in a form relevant tocoastal oceanography, that is, for a model where 52s w

(z, t) and where a mean W exists (e.g., tidal regime),2s w

the full solution from Thomson with w 5 W 1 w is

2 2 2w 1 ]s ]W 1 ]s ]sw w wdw 5 2 1 1 1 1 W w21 2[ T 2 ]z ]t 2s ]t ]zL w

2 21 ]s 2sw w21 w dt 1 3 zÏdt ,2 ] !2s ]z Tw L

(4)

where TL is the Lagrangian timescale, and z is aÏdtrandom number from a Gaussian distribution with var-iance dt. A Lagrangian stochastic model is consideredvalid for timescales smaller than TL but longer than theKolmogorov timescale th 5 (n/e)1/2, where n is the ki-nematic viscosity and e is the turbulent kinetic energydissipation. A more familiar form for (4) is with ] /2s w

]t 5 0 (i.e., stationary turbulence) and no mean W:

2 2w 1 ]s wwdw 5 2 1 1 1 dt21 2[ ]T 2 ]z sL w

22sw1 3 zÏdt . (5)! TL

This is the equation derived by Wilson et al. (1981).Legg and Raupach (1982) derived a similar equationbut without the w2/ term (therefore not a well-mixed2s w

model). (The difference between the Wilson et al. andLegg and Raupach models is relevant in highly inho-mogeneous turbulence.) The interpretation of (5) is thatspatial gradients in the magnitude of turbulent kicks (therandom term) lead to accumulation of particles in re-gions of low turbulence, a tendency that is counteredby the ] /]z factor in the drift term.2s w

c. The inhomogeneity index in 1D turbulence

If we nondimensionalize (5) using

w 5 s w9 t 5 T t9,w L

we get (dropping primes)

2dw 5 {2w 1 I (w 1 1)}dt 1 zÏ2dt , (6)H

where

2T dsL wI 5H 2s dzw

is the inhomogeneity index [as mentioned in the Wilsonand Sawford (1996) review], and in the sense that theseequations are Lagrangian equations, z 5 z(t) so that IH

5 IH(z(t)). Note that in the context of integrating the1D model, IH depends on the input data. Although theequations are typically not integrated in nondimensionalform, the inhomogeneity index serves as a useful in-dicator for possible problems with the input data. (Note:IH as a measure of turbulence inhomogeneity can bemisleading because, if ] /]z 5 0 and TL 5 TL(z), then2s w

IH 5 0, but the turbulence is still considered inhomo-geneous. This is often the case in the lower atmosphere.)

d. The 3D solution is not unique

The solution of the 3D Langevin equations is con-siderably more complicated and, importantly, no uniquesolution is known to exist. (The same is true in 2D.)Thomson derived his ‘‘simplest’’ solution, in which eachvelocity component is of the form

2du 5 f (U , u , s , x, y, z, t, u u , . . .), (7)i i i i i j

containing 74 terms. Note that for the typical coastalcirculation model where, for example, ; K]U/]y,u9y9the terms in (7) are derivable. To the authors’ knowledgeno attempt has been made to use a 3D LSM of thisnature. If 3D stochastic modeling is required, an RDM,described below, is preferable. A final problem with thesimplest solution is that it is impossible to determine ifit is a physically realistic solution. In the atmosphere,the 3D model above is often used in its 2D form (Rodean1996, p. 44; Flesch et al. 1995) after simplifying as-sumptions are made regarding the mean flow field andturbulent correlations ( etc.).u9y9


e. The random displacement model

A random displacement model is the ‘‘diffusionlimit’’ of an LSM: the limit TL → 0 or t k TL. Thesemodels of particle displacements get around the prob-lems with the 3D (and 2D) LSMs. Various derivationsof this limit for the 1D model exist in the literature (see,e.g., Durbin 1984; Boughton et al. 1987). Here we quotethe general result from Rodean (1996):

]K (x, t)ij 1/2dx 5 U (x, t) 1 dt 1 [2K (x, t)] dW (t), (8)i i ij j[ ]]xj

where Kij is the eddy diffusivity. Equation (8) is aunique solution to the turbulent diffusion problem. Ap-plying the Fokker–Planck equation, rearranging andusing the fact that the concentration C is related to thepdf P leads to

]C ]C ] ]C1 U 5 K , (9)i ij1 2]t ]x ]x ]xi i j

which shows that the RDM is equivalent to an advec-tion–diffusion equation model. We see that the RDMprovides a simple, unique, 3D alternative to an LSM,valid for timescales much longer than TL.

f. TL and the accuracy of an LSM versus an RDM

In general, an LSM is considered to be more accuratefor short timescales and space scales and in complexgeometries than an advection/diffusion equation model(or RDM; Du et al. 1994). The reason for this can beunderstood using Taylor’s classic theory of turbulentdispersion (Taylor 1921). First, we define the Lagrang-ian timescale as the integral of the velocity autocorre-lation function r:

`

T 5 r(t) dt.L E0

[Note that in the context of LSMs TL 5 (2 )/(C0e)2s u

(Wilson and Sawford 1996), which for steady homo-geneous turbulence is an integral timescale. For un-steady, inhomogeneous turbulence the definition still ap-plies locally.] Taylor’s theory for the evolution of theensemble average variance X 2 is

t21 d^X &25 s r(t) dt, (10)u E2 dt 0

where we think of the left-hand side as equal to theeffective diffusivity K, and the autocorrelation functionon the right-hand side to have the classic decaying ‘‘ex-ponential correlogram’’ form. For timescales KTL whenthe initial patch is much smaller than the turbulent eddyscale, we get

2K ø s t,u

while in the diffusion limit, when the patch has grownlarger than the eddy scale, we get a K-theory model

2K ø s T .u L

Thus K-theory, or eddy diffusion, models are consideredto overestimate dispersion for times much shorter thanTL and to be inaccurate near source regions. As well,in complicated terrains it is considered more accurateto track particles than to solve an advection–diffusionequation.

The above conclusions must be tempered by twofacts. First, the velocity and turbulent quantities drivingan LSM are usually output from a discretized circulationmodel, which introduces separate questions of accuracy.Second, it typically takes 10–100 times more computertime to integrate an LSM versus an advection–diffusionmodel, and this must be weighed when considering the(possible) convenience and added precision of an LSM.

g. Boundary effects

An LSM must be complemented with a correctboundary reflection scheme in order to preserve parti-cles numbers (no flux condition) and velocity correla-tions, and to allow the WMC to be satisfied. The detailsare complicated (Wilson and Flesch 1993) but can bedistilled down to the requirement that the turbulencemust become homogeneous as a boundary is ap-proached. As there is always an unresolved boundarylayer, this condition can be artificially imposed on thedata. In practice, however, correct results are often re-ported in violation of these theoretical requirements(e.g., Legg and Raupach 1982).

h. Time step choice and dt bias

Typically, LSMs use an explicit time-stepping schemefor the stochastic component. In inhomogeneous tur-bulence where TL changes with position, it is temptingto use an adaptive scheme whereby the time step chang-es based on the local TL, instead of one in which dt ,min(TL). Wilson and Flesch (1993) showed that an adap-tive stepping scheme can result in what they call a dt-bias effect. Briefly put, they considered a variable timestep approach in inhomogeneous turbulence character-ized by 5 const and TL 5 TL(z) and showed that2s w

this leads to a net bias velocity due to a particle in a TL

gradient having a different dt depending on whether itwas going up or down. This bias velocity can lead tothe unmixing of initially uniform particle fields. There-fore, it is safer to use a constant dt.

i. Nonneutrally buoyant particles

Up to this point, it has been implicit in our discussionthat particles are neutrally buoyant and act like fluidparcels (the WMC required this). In reality, many par-ticles of interest (e.g., fish eggs or larvae) are not neu-trally buoyant or may exhibit deliberate behaviors (e.g.,swimming). Unfortunately, a satisfactory theory forLSMs of nonpassive particles does not exist.


The essence of the problem is the following. Imaginetwo time steps of an LSM in which the particle (an egg)has some density-dependent behavior (e.g., Stoke’s set-tling velocity). On the first time step the fluid particlemoves from z0 to z1 where it has velocity w1. The eggmoves to z1 and then to due to its settling velocity.z91The trouble begins on the second time step where wehave a fluid particle at z1 with w1 and an LSM to describeits next step (from z1), but the egg is at with a differentz91w. This is called a trajectory crossing problem. Solutionsin the literature (Sawford and Guest 1991; Zhuang etal. 1989) consider the spatial correlation between thefluid particle and the ‘‘egg’’ and adjust TL or reset theeddy that the egg is in, depending on assumptions re-garding this correlation. The models work fairly wellbut indicate that much work remains to be done. Foroceanographic purposes, where egg–water density dif-ferences are typically small, the trajectory crossingproblem may not be that important (provided the runlength is short enough for this to be true). But if sig-nificant swimming behavior is involved, then the chanc-es of a particle wandering out of its eddy may not besmall and significant tracking errors may occur.

3. Demonstrating the WMC

In this section we look at the ability of a 1D LSMand a 2D RDM to demonstrate the well-mixed conditionusing circulation model turbulence data as input. Wewill see that the output turbulent data from circulationmodels can make demonstration of the WMC impos-sible, regardless of time step or the number of particlesused. We speculate that this is due to poor representationof the spatial derivatives of turbulent quantities requiredby the stochastic model. We show for smoothed modeldata and discretized analytical turbulence data that byincreasing the output resolution the WMC can ulti-mately be demonstrated.

a. Input data

The turbulence data used in this paper is output froma run of the regional finite element model Quoddy4(Lynch et al. 1996). The circulation model domain ex-tended from the eastern Nova Scotian Shelf to west ofGeorges Bank and from about 150 km offshelf to thecoast. The model had 21 vertical sigma levels designedwith higher resolution in both top and bottom boundarylayers, 12 192 nodes, and included density, wind stressand tidal forcing.

Vertical turbulence variables q2, q2l, kq (turbulent ki-netic energy, length scale, and vertical diffusivity) arecalculated from a Mellor–Yamada 2.5-level scheme(Mellor and Yamada 1974). The derived quantities foruse in an LSM ( 5 0.3q2/2, TL 5 kq/ ) are output2 2s sw w

at the sigma coordinate levels. Horizontal turbulence isrepresented by a horizontally (and temporally) varyingdiffusivity AH calculated using a Smagorinsky (1963)

scheme. (Note that AH does not vary in the vertical.)For use in the 2D RDM tests described below, the datawere interpolated onto a 5 3 5 km grid. For all exper-iments reported in this paper we use tidally averaged(residual) turbulent quantities.

Figure 1 (thick lines) shows a vertical profile of tur-bulence data from a grid point near Browns Bank. Thedata, with strong vertical variation in and TL, are2s w

considered to be highly inhomogeneous. An inspectionof the dataset showed that approximately 85% of thenodes had profiles of this nature. Note that the turbu-lence, represented by ] /]z and IH, only weakly ap-2s w

proximates a homogeneous structure at the boundariesand thus does not satisfy the Wilson and Flesch (1993)boundary condition requirements. For the 1D tests de-scribed below, the data were artificially modified toforce homogeneity at the boundaries. In practice thismade no difference to the results.

Figure 2a is a contour plot of the AH field over BrownsBank. The diffusivity ranges from 50 to 200 m2 s21,with generally lower values on-bank. For the 2D test,as in the 1D case, gradients in AH were forced to zeroat the boundaries.

b. Methods for demonstrating the WMC

One run of an LSM can be considered to be one trial,or sample, of the possible ensemble of trials, and anymethod for demonstrating the WMC, or assessing theresults from an LSM run, has to account for the statis-tical nature of the output of an LSM. The demonstrationof the WMC requires showing that the initial uniformparticle concentration (or distribution) remains uniform,in a statistical sense. Due to the nature of the inputturbulent data, we use a different method for the 1Dand 2D cases.

1) THE 1D CASE: THE VARIANCE TEST

In one dimension we test for the WMC as follows.Multiple trials of an LSM are performed starting witha uniform distribution of P particles. For each (24 h)trial the time mean concentration

[ (z)] at N equally spaced levels is calculated. [Con-Ccentration is calculated using a residence time algorithm(Luhar and Rao 1993).] For an ensemble of T trials theensemble mean concentration ^ & 6 std dev is com-Cputed. If ^ & is within one std dev of the initial con-Ccentration at all levels, then we consider the WMC tobe demonstrated. However, this fact alone is not suffi-cient because we need to know something about thestatistical stability of this result, that is, whether or notthis is the expected result or just a chance occurrence.

We determine this by repeatedly subsampling the totalM-trial ensemble and forming the statistics


FIG. 1. Circulation model turbulence data 2 , ] /]z, TL, IH 5 TL/ ] /]z, from near Browns Bank. Thin dashed lines are polynomial2 2 2 2s s s sw w w w

fits to the data. Symbols mark data positions.

N T

2 2S 5 (1/R)(1/N ) (C 2 ^C & ) /(T 2 1) (11)O O ij iji j

and2S

V 5 , (12)T

where S 2 is the mean of the distribution of sample var-iances, (V)1/2 is the standard deviation of the mean orstandard error, N is the number of levels, T the numberof trials in the sample, and R the number of independentT-sized samples drawn from the M-trial ensemble (Steeland Torrie 1960, p. 56). We expect that V(T) will be adecaying function of sample size T, and the decay scalein T gives an indication of the number of trials (orparticles) required to yield a statistically acceptable re-sult. We call this procedure the variance test.

2) THE 2D CASE: THE CORRELATION TEST

In 2D it is more complicated to devise a WMC test.Our test is based on the fact that if a WMC does notexist, then a significant correlation would exist betweenthe perturbation concentration field C9(x, y, t) 5 C(x,y, t) 2 and the perturbation diffusivity field (x, y)C A9H5 AH(x, y) 2 , where is the initial well-mixedA CH

concentration and is the xy-averaged diffusivity field.AH

To make this idea clearer, Figs. 2a,b show the BrownsBank AH field and the concentration field at day 16 from

a run in which the drift (i.e., =AH) term in the 2D RDM[Eq. (8)] is turned off. The negative C9: correlationA9His obvious with regions of higher concentration corre-sponding to lower diffusivity and vice versa.

To test for the WMC, we calculate the normalizedcorrelation coefficient versus time,

C9 · A9HCor(t) 5 , (13)2 2Ïs sC9 A9H

where C9 · 5 Si S j , and s2 is the variance.A9 C9A9H ij Hij

Significance levels are calculated using a version of thetechnique of Perry and Smith (1994). At a given time,the C9 field was repeatedly randomized and the teststatistic Cor recalculated. A histogram of the 2000 Corvalues was constructed and 6Cor values correspondingto 90% and 95% of the area was determined. So, forexample, the 95% significance values mean that giventhe set of C9s at a particular time, 95% of the time wewould expect a random correlation between C9 and

to lie between 6Cor95. If the actual Cor is outsideA9Hof this range, then it is deemed significant; and if thisresult pertains for all time steps and numbers of parti-cles, then we consider that the WMC is not attained.

c. The 1D LSM tests

Using the ‘‘raw’’ model data (Fig. 1) the WMC wasnever found to be met. This is illustrated in Fig. 3, whichshows a plot of ^ & 6 std dev, averaged over ten 500-C


FIG. 2. (a) Browns Bank AH field; (b) concentration field at day 16 of an incorrect 2D RDM.Darker shading denotes higher values. Note the negative correlation between concentration andAH. Thick contours are isobaths at 75 (dashed line), 100 (solid line), and 200 m (dotted line). Ticmarks are at 10-km spacing.

particle trials. The ensemble mean concentration haslarge dips between 210 and 220 m, and 240 and 250m, indicating that particles are evacuated from that zone.These problem regions correspond to regions of rapidlychanging IH (Fig. 1). The plot also shows the behaviorof V versus the number of trials, determined by re-peatedly subsampling the 20-trial ensemble. Note thatat least 10 trials are required before V levels off. Theidea that about 5000 particles are required to achievestatistical certainty in an LS model is consistent withthe atmospheric literature on this subject. The aboveresult pertained regardless of the time step chosen orthe method of calculating vertical derivatives. As well,runs where the boundaries were extended ‘‘to infinity’’with homogeneous turbulence—to eliminate possibleboundary effects—showed no difference.

Two possible reasons for failure to meet the WMCare as follows. 1) The model formulation is somehowdeficient (e.g., due to truncation at some lower order)and higher-order derivative terms are required to makethe model work in highly inhomogeneous turbulence.Or 2) the input data resolution is such that calculationof derivatives is too imprecise to allow demonstrationof the WMC; that is, the data are effectively discontin-

uous. In the derivation of LS models (Gardiner 1983;Rodean 1996) it is not obvious that important higher-order terms are omitted, and this fact plus the historicalsuccess in the use of LSMs makes us doubt that themodel formulation is deficient. Given the nature of theinput data (Fig. 1) it is more likely that the resolutionoffered by the raw data combined with the inhomoge-neous nature of the turbulence leads to fields that arenot resolved smoothly enough to allow the model tosatisfy the WMC.

To demonstrate this, the TL and data were fit to2s w

sixth- and fifth-order polynomials, respectively, and out-put at 0.5-m resolution (Fig. 1, dashed lines). Note thatthe fit produces smooth and reasonable looking and2s w

TL profiles and significantly reduces the amplitude ofthe IH profile. Figure 4a shows ^ & 6 std dev (10 trials)Cfor the smoothed-data model. The mean curve has lotsof small-scale wiggles but is within one std dev fromfive particles per meter at all levels, thus satisfying theWMC. Figures 4b,c show the behavior of the averagevariance and the fraction of trials meeting the WMC asa function of number of trials. The latter was computedby repeatedly subsampling the 20-trial ensemble andshows again that at least 10 trials must be done (5000


FIG. 3. (a) Ensemble mean 6 std dev for 10 trials; (b) result fromvariance test.

FIG. 4. (a) Ensemble mean concentration field 6 std dev, plus initialconcentration line; (b) result from variance test; (c) fraction of trialsmeeting WMC vs number of trials in sample.

total particles) to have statistical confidence in the LSMoutput.

The above runs were done using a constant time stepof 0.5 s (almost 20 times smaller than the minimum TL

in the dataset). Figure 5 is a representative result for dt5 10.0 s, showing an evacuation of particles near 240m as in Fig. 3 and problems near the bottom boundarywhere the time step is O(TL). It was found that if thetime step is increased to 10 s, the WMC is never metfor the same input data. Therefore, even for smooth dataan incorrect choice of time step can lead to incorrectresults.

To test the hypothesis that data smoothness–resolu-tion, with its effect on derivative calculation, is the rea-son that the WMC is not met, we used Gaussian ana-lytical TL and profiles2s w

22(z /10)T 5 100.0e 1 10.0L

22 24 2(z /10) 24s 5 3 3 10 [1 2 e ] 1 1.5 3 10 ,w

output at increasing vertical resolution, and investigatedwhether the WMC could be satisfied. The reason for thechoice of a Gaussian shape is that it roughly represents

the vertical profile of turbulence in a two–boundary lay-er system and has the properties of strong variation inIH plus homogeneous turbulence at the boundaries, thusremoving the possibility that problems are due to bound-ary effects.

Figure 6 shows the result for turbulence data outputat 2.5-m resolution, for which the WMC was not met.Figure 6a contains the 20-trial ensemble of curves, ex-hibiting a peak at zero depth with only one curve havinga value on the low side of the eight particles per metermean concentration near that depth. This is illustratedin Fig. 6b where the 20-trial ensemble mean 61 std devand the mean concentration line are plotted. Figure 6cshows the average variance versus number of trials in-dicating that at about 10 trials a statistical stable resultis obtained.

Figure 7 shows the result for turbulence data outputat 0.25-m resolution where the WMC is attained (Fig.7a). Figure 7b shows the variance versus number oftrials, again leveling off around 10 trials. Figure 7cshows the fraction of trials satisfying the WMC versusnumber of trials indicating that 16 trials (8000 particles)


FIG. 5. Ensemble mean concentration field 6 std dev, plus initialconcentration line for 10-s time step. The WMC was not met for thiscase.

FIG. 7. Result from 1D LSM using Gaussian turbulence data outputat 0.25-m resolution. In this case the WMC is demonstrated.

FIG. 6. Result from 1D LSM using Gaussian turbulence dataoutput at 2.5-m resolution. The WMC was not met.

are required to demonstrate the WMC with 100% cer-tainty.

We note that the above results are sensitive to thepeak-to-peak amplitude of IH(DIH), a tunable parameter.As DIH decreases it gets progressively easier to dem-onstrate the WMC; as it increases a point is reachedwhere it is impossible to show a WMC except whenanalytical functions are used within the LSM to cal-culate derivative quantities.

The above results support the hypothesis that reso-lution problems and their effect on turbulent derivativecalculation lead to the inability to satisfy the WMC.

d. The 2D RDM tests

In this section we look at the performance of a 2Drandom displacement model [Eq. (8)] using circulationmodel output data (AH field) and an analytical AH fieldas input data. Figure 8 is a plot of the normalized cor-relation coefficient versus time, and the 95% signifi-cance level, for a run using the circulation model AH

field. The figure shows that the correlation levels offafter about 18 days and is significant; that is, the WMCis not met using the AH field output from the circulation


FIG. 8. Correlation ^C9 · & vs time plus 95% significance level9AH

for a WMC test using the Browns Bank AH field. The WMC is notmet.

FIG. 9. (a) Correlation ^C9 · & vs time plus 95% significance level9AH

for a WMC test using a 2D Gaussian AH field output at 10-km res-olution showing that the WMC is not met; (b) same as (a) exceptthat the 2D Gaussian AH field is output at 2-km resolution and themore stringent 90% significance interval is plotted. In this case theWMC is demonstrated.

model. The leveling off of the correlation represents thetime it takes for a diffusive equilibrium to be estab-lished. Changes in the time step and number of particlesmade no difference to this result.

To test whether the inability to demonstrate a WMCwas due to resolution effects we ran the RDM using a2D symmetric analytical Gaussian AH field, output at 2-and 10-km resolution. The origin of the Gaussian wasat the center of the domain with a spread of 20 km. Thecorrelation plots for 10- and 2-km resolutions (Fig. 9)show that, as in the 1D case, a finer resolution allowsthe WMC to be realized. As in the 1D case, if the ‘‘in-homogeneity’’ of the 2D AH field is increased (by de-creasing the spread), then the demonstration of theWMC becomes more difficult unless the analyticalfunction is used to calculate the spatial derivatives.

To summarize, circulation model output turbulencedata in both the vertical and horizontal directions maybe such that the well-mixed condition cannot be dem-onstrated. Tests using analytical forms of turbulencedata show that this is due to resolution problems withinhomogeneous turbulence data. As output resolution isincreased, derivatives of turbulent quantities are betterdetermined and the WMC can be demonstrated.

4. Determining the number of particles

The tests for the WMC illustrate the principle methodfor determining whether enough particles were used:perform multiple trials and investigate how the answer(or some measure of it) changes. The variance test isan example of this procedure. Without a test of thisnature one cannot be certain that the output from the

stochastic model is representative of the (desired) en-semble statistics, or an outlier from which it would bedangerous to draw conclusions. In this section we iden-tify two types of errors due to underseeding (or under-sampling) the source region and show that the numberof particles required, and thus the potential for error, isspecific to the problem under consideration.

To illustrate what we will call a U-I underseedingerror (U-I error) we ran a 1D LSM on the simple prob-lem of relaxation of an isolated uniform plug of tracerin homogeneous turbulence (the ‘‘top-hat’’ problem).The analytical solution to this problem provides aground truth to evaluate the LSM performance. Figure10a shows the concentration profile after 3 h for ten500-particle trials of an LSM, with initial uniform seed-ing between 612.5 m. Figure 10b shows the ensemblemean 6 std dev for the LSM and the analytical solution.Note that the ensemble average accurately simulates theanalytical solution, and that any of the 10 trials wouldreasonably represent the analytical concentration pro-file. That is, 500-particle seeding is dense enough thata single trial is representative of the ensemble and thusapproximates the correct answer everywhere.

Figure 10c is the same as Fig. 10a, except that the


FIG. 10. (a) Ensemble of 10 trials of a 500-particle LSM for the relaxation of an initial uniform concentration of tracer (time 5 3 h); (b)the ensemble mean concentration (thick line) 6 std dev (dashed lines), plus the analytical solution (thin line); (c) same as (a) except thatthe LSM is seeded with 50 particles; (d) same as (b), for 50-particle trials.


FIG. 11. Particle dispersal at 10, 20, 30, and 40 days, for a 500-particle trial. The particles are released on Browns Bank. The figureshows the location of the bank, plus a 20 3 20 km square downstream from the release site. Tic marks are at 50-km spacing.

initial domain is seeded with 50 particles. In this case,the variation in the 10 trials is much larger, althoughthe ensemble mean plus deviation (Fig. 10d) still cap-tures the analytical solution. However, if one were tohave done just the one trial highlighted by the widerblack line in Fig. 10c, errors as great as 100% (compared

to the analytical solution) would have resulted. There-fore, if the source region is underseeded, an individualtrial can poorly represent the ensemble and thus lead toerroneous conclusions. We call this a U-I error.

To illustrate a U-II error and show how the requirednumber of particles is specific to the problem under


FIG. 12. The fraction of particles retained within the 100-m isobathon Browns Bank for 100, 500, 1000, and 2000 particles.

FIG. 13. The fraction of particles found in a 20 3 20 km squarenear the mouth of the BoF as a function of time for sixteen 2000-particle trials.

consideration, we refer to more relevant examples usingQuoddy flow fields for the southwest Nova Scotia–Bayof Fundy (SWN–BoF) region.

Browns Bank is the principal spawning ground forhaddock in the SWN-BoF ecosystem, and there is con-siderable interest in the partitioning of haddock eggsand larvae between the bank, with its retaining gyre,and downstream in the BoF (Campana et al. 1989;Shackell et al. 1999). We used the Quoddy 3D clima-tological spring flow field with residual horizontal tur-bulence (AH field) to track particles spawned on thesoutheast portion of the bank to look at two questions:1) what is the fraction of particles retained on the bankas a function of time, and 2) what is the concentrationin some downstream grid cell as a function of time? Wedesire to know the number of particles/trials requiredto answer these questions with statistical confidence andaccuracy.

As an illustration of the problem, Fig. 11 shows par-ticle drift and dispersion at 10-day intervals for a 500-particle trial. The Browns Bank 100-m isobath is out-lined, plus a 20 3 20 km box downstream at the mouthof the BoF. In this experiment, particles are tracked atthe 20-m level, and the AH field is smoothed in orderto avoid problems with the WMC.

Figure 12 shows the fraction of particles retainedwithin the Browns Bank 100-m isobath as a function oftime, for various total number of particles (100–2000).All of the curves are remarkably similar, with as fewas 100 particles (solid line) yielding an answer accurateto about 10% after 20 days. For this question, 500 par-ticles are all that are required to obtain a correct answer.

To understand why this is so, think of this as a source-target problem, with Browns Bank as the source and,effectively, Browns Bank (or not Browns Bank) as thetarget. Looked at in this way, it is logical that few par-ticles are required for an accurate answer because thetarget area is large and is thus ‘‘easy to hit.’’

Looked at in the same way, we can anticipate thatthe problem of accurately determining the concentrationat a remote downstream location will be more difficultand will require more particles. Figure 13 shows thefraction of particles found within a 20 3 20 km squarenear the mouth of the BoF at days 45–60 for sixteen2000-particle trials. Note first that the maximum spreadis about a factor of 2 at day 50, and by eyeballing themean and scatter at each time, one would estimate thata 2000-particle trial is within about 30%–40% of theensemble mean. If we were concerned with the tem-perature field that the particles experienced, and thisfield changed over a scale of hundreds of kilometers,then this level of accuracy could be considered goodenough. If, on the other hand, the particles representedan oil spill, and the 20 3 20 km grid cell part of amajor feeding ground, then greater accuracy may berequired.

To get a more precise measure we use the variancetest described in section 3. Figures 14a,b show the meanconcentration and std error of the mean at day 45 as afunction of number of 2000-particle trials, determined(as above) by repeatedly subsampling the 16-trial en-semble. Figure 14c shows the 95% confidence intervaldetermined from a Student’s t-test, expressed as a frac-tion of the mean. We see that the mean and std error


FIG. 14. (a) Sample mean; (b) std dev of the mean; and (c) 95%confidence interval for the sixteen 2000-particle trials (day 5 45).

FIG. 15. Sample mean, std dev of the mean, and 95% confidenceinterval for twelve 500-particle trials (day 5 45).

stabilize at around eight trials (or 16 000 particles), atwhich point the t-test would indicate that the true meanwould lie between 610% of the estimated mean(ø0.0325), at the 95% confidence level. Note that themean varies only slightly with the number of trials, andthe std error is small (;0.002/0.0325 ; 6% or less)even for two trials, or 4000 particles. Therefore, withless confidence, but also considerably less computertime, one 4000-particle run would give a satisfactoryanswer.

The same technique can be used on 500-particle trials,shown in Fig. 15. The figure shows that again the meanand standard error stabilize around 9–10 trials, but witha much higher std error (ø25%) and a confidence in-terval of about 60% of the mean. Therefore, 10 3 500particles does not seem to yield a satisfactory result andcalls into question the implied equivalence between Ntrials of M particles and one trial of N 3 M particles.Note, however, that the mean concentration is roughlyhalf as large as that seen in 2000-particle runs. Thisillustrates what we call a U-II error, resulting from un-derseeding the source area in a way that does not suf-ficiently sample the subarea that would, on average,have trajectories that pass through the target. This prob-lem, which also occurs for strictly deterministic flowfields, is illustrated in Fig. 16. Consider the situation,for deterministic flow, in which the middle 20% of asource region (S9 5 0.2S) is the source for trajectories

that intercept the target region (T) downstream (Fig.16a). Let us uniformly seed the entire region with anincreasing number of particles and predict the fractionof particles that will intercept the target. One particleseeded uniformly in S would lie in S9 and thus we wouldpredict that 100% of particles would hit T. Two particleswould miss S9 completely and lead to a 0% prediction,et cetera. Figure 16b shows the prediction as a functionof the number of uniformly seeded particles and indi-cates that there is a minimum number of particles re-quired to obtain an accurate answer. Figure 16c showsthe day-45 concentration of particles in the 20- 3 20-km grid discussed above for 100-, 500-, 1000-, 2000-,and 4000-particle trials and shows that for this specificproblem at least 1000 particles are required to avoid aU-II error.

5. Discussion

In this paper we have presented various aspects ofLagrangian stochastic modeling directed toward prac-tical applications in coastal oceanography. Starting froma general review of theory, we looked at one- and two-dimensional tests of the well-mixed condition and theproblem of determining the number of particles or trialsrequired to produce reliable results.

Oceanographic stochastic modeling tends to be dif-ferent from atmospheric modeling in that turbulent data


FIG. 16. (a) Schematic of a source region S with a subregion S9whose trajectories intercept the downstream target, and (b) the pre-dicted fraction of particles hitting the target vs the number of particlesuniformly seeded in S; (c) concentration vs number of particles forparticles seeded on Browns Bank and intercepting the 20 3 20 kmsquare at the mouth of the BoF 45 days later.

FIG. 17. Comparison of DM, RDM, and LSM using the input tur-bulence data shown in Fig. 1. (a) DM solution at 24 h; the thin dashedline is the initial concentration, the thin solid line is the ky profile.(b) Ratio of RDM/DM and LSM/DM. Plotted are the ensemble meansof 20 500-particle trials. The std devs are ,5% and are omitted forclarity.

for use in oceanographic stochastic models are outputfrom ocean circulation models while atmospheric mod-els use a surface stress and analytical functions to pro-vide vertical structure. We found (section 3) that theoutput turbulence data from coastal circulation modelscan be effectively discontinuous so that first derivativesdo not exist, and that this makes demonstration of theWMC impossible. We showed that this was likely thecorrect interpretation by using analytical turbulence dataoutput at various resolutions and showing that the WMCcould only be demonstrated for sufficiently high reso-lutions. In all cases discussed, the use of the analyticalfunctions themselves removed this problem. This makesthe case for the development of canonical vertical struc-ture functions for oceanic turbulence. Although beyondthe scope of this work, such solutions may be possiblefrom simplification of the governing turbulence closureequations. Loss of accuracy due to simplification couldbe balanced against the ability to use vertical turbulentschemes in oceanographic particle tracking.

Despite difficulties in demonstrating the WMC, it ispossible that for practical problems acceptable answerscould still be obtained; that is, the WMC test could betoo rigorous. It is difficult to answer this question be-cause the alternative to an LSM, an advection–diffusionequation model, suffers from inaccuracies as well, prin-cipally involving artificial diffusion, and overestimationof dispersion for short release times (see, e.g., Zam-

bianchi and Griffa 1994). Nevertheless, we compared a1D RDM, a 1D LSM, and a diffusion equation model(DM) based on the turbulence data in Fig. 1 for theproblem of dispersion of an initial concentration of pas-sive tracer with a peak at 230 m. [This profile is meantto simulate the observed distribution of haddock larvaein SWN. See Frank et al. (1989, Fig. 4.)] Despite ar-guments about the degree of diffusion in the diffusionmodel, the pattern obtained will be essentially correct.Figure 17a shows the DM solution at 24 h, plus theinitial concentration and input ky profiles. The errormeasures ratio RDM/DM and LSM/DM are plotted inFig. 17b. We see that the greatest LSM errors occur atexactly the regions where the WMC problems occurred(Fig. 3a) and that the RDM arguably does slightly betterthan the LSM. This comparison indicates that if thereare problems in demonstrating the WMC, then this is awarning flag for the use of a vertical turbulent schemein a particle tracking routine.

We discussed (section 4) how to determine the num-


ber of particles, or number of trials, required to havestatistical confidence in one’s results. The method in-volves doing repeated trials using the stochastic model,forming the ensemble average and standard deviation,and determining when these statistics stabilize. Thismethod was used to determine whether the WMC wassatisfied (the variance and correlation tests described insection 3).

We identified two types of errors related to under-seeding the particle source region (section 4). What wecall a U-I error was illustrated by using an LSM to modela problem with an analytical solution. This type of erroris due to using so few particles that significant deviationfrom the desired answer may result. In other words theuncertainty in the output is so large that the result maybear little resemblance to the true answer. We see thisas akin to an undersampling problem in which the un-derlying distribution is so poorly sampled that the sta-tistics of the sample do not properly represent the sta-tistics of the distribution.

A U-II error is also an undersampling error, but itinvolves an underseeding of the part of the source regionwhose trajectories intersect the target area and illustratespossible problems associated with indiscriminate use ofstatistical tests. Essentially, it is possible with sparseseeding to miss source regions whose (turbulent)streamlines intersect the desired target. Repeated trialswith the same seeding lead to stable but incorrect sta-tistics. The example we gave (section 4) showed howone could underpredict downstream particle concentra-tion by a factor of 2.

In Lagrangian stochastic modeling there is an appar-ent equivalence between N trials of M particles, and onetrial of N 3 M particles. In the example used to illustratea U-I error, we saw that ten 50-particle trials or one500-particle trial gave acceptable results. However, theU-II error example pointed out a caveat in this equiv-alence: if a U-II underseeding situation pertains, theensemble of any number of trials will give the wronganswer. The lesson is that one has to test for a minimumnumber of particles as well as statistical stability (Fig.16).

Associated with the above are the ideas of how thequestion being asked determines the required numberof particles and what a satisfactory answer is. The ex-ample of the fraction of particles seeded on BrownsBank that were retained on Browns Bank as a functionof time required few particles (;500) to achieve anaccurate answer, while the accurate determination of theconcentration in a 20 3 20 km region downstream atthe mouth of the Bay of Fundy required a minimum ofeight 2000-particle trials. In the latter example weshowed that the uncertainty, once the minimum numberof particles was established, was about 30% so that ifaccuracy requires are low, one 2000-particle trial wouldbe adequate (Fig. 13).

We discussed the variability in the required numberof particles in terms of a source–target problem. If the

target is ‘‘large’’ and easy to hit, as in the case of theretention problem, then few particles will be required.If the target is ‘‘small,’’ then more particles will berequired and the possibility of a U-II error increases. Itis difficult to make the concept of effective target sizerigorous, but factors involved would be the distributionof streamlines emanating from the source that intersectthe target, the turbulent intensity between the sourceand target, and the timescale for traveling the source-target distance. Because Lagrangian stochastic modelingis a computer-intensive task with repeated trials of largenumbers of particles typically necessary to produce be-lievable results, a recipe based on the source–target con-cept would be a useful tool to guide modeling efforts.

The increase in the use of Lagrangian stochastic mod-els in coastal oceanography is driven by the use of in-dividual-based models in biophysical modeling of thedispersion, growth and mortality of fish eggs and larvae.The main advantage of individual-based models is thesimple representation of growth and mortality functions.Although problems exist in translating these relation-ships to continuum mechanics form, doing so allowsthe complete model to be recast in advection–diffusionequation form. This transformation would eliminateproblems related to the well-mixed condition and speedup computation time. It is possible that the work re-quired to convert individual-based models to concen-tration-based equations will be well worth the effort.

Acknowledgments. Author D. Brickman was sup-ported by U.S. GLOBEC.

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