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CFD application to oceanic mixed layer sampling withLagrangian platformsTamay M. Özgökmen a & Paul F. Fischer ba Rosenstiel School of Marine and Atmospheric Science, University of Miami, Florida, USAb Mathematics and Computer Science Division, Argonne National Laboratory, Illinois, USA

Available online: 21 May 2012

To cite this article: Tamay M. Özgökmen & Paul F. Fischer (2012): CFD application to oceanic mixed layer sampling withLagrangian platforms, International Journal of Computational Fluid Dynamics, DOI:10.1080/10618562.2012.668888

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CFD application to oceanic mixed layer sampling with Lagrangian platforms

Tamay M. Ozgokmena* and Paul F. Fischerb

aRosenstiel School of Marine and Atmospheric Science, University of Miami, Florida, USA; bMathematics and Computer ScienceDivision, Argonne National Laboratory, Illinois, USA

(Received 16 June 2011; final version received 9 February 2012)

Frontal adjustment and restratification in oceanic mixed layers is one of the processes that is considered to beimportant in the ocean’s multi-scale energy transfer, biogeochemical transport, air–sea interaction, acousticpropagation and naval operations. We summarise a CFD-based modelling approach to sample processes at anidealised mixed layer base using passive scalars and particles, given a subset of realistic constraints on these resourcesin field experiments. The results emphasise the effectiveness of Lagrangian platforms, in particular passive particles,for sampling rapidly evolving submesoscale oceanic fields.

Keywords: ocean mixing; ocean sampling; Lagrangian transport; tracers; relative dispersion

1. Introduction

1.1. Importance

Oceanic mixed layers are important for several reasons.First, themost pronounced and energetic eddies and jetsin the ocean circulation have a scale on the order of theRossby deformation scale, O(104)m (so-called mesos-cale range), while viscous dissipation of kinetic energytakes place at scales on theO(1072)m. Since the ocean isa forced-dissipative system, it is important to know howthe energy driving the ocean circulation is dissipated, forreliable forecasts of the ocean state, and the climate. Thepotential routes to dissipation can be classified intothree (Muller et al. 2005): (i) the ocean’s internal gravitywaves can interact with mesoscale motions and catalysea cascade of their energy to dissipation scales; (ii)balanced mesoscale flows can become unstable andinitiate a down-scale energy cascade and (iii) dissipationcan take place by flow interactions with complex, roughbathymetry. Little is known about processes in theintermediate, or submesoscale range (Wunsch andFerrari 2004). The submesoscale range is generallycharacterised by coherent features in the spatial scalerange of 102 m to 104 m and an eddy turn over time scaleof the order of days (Thomas et al. 2008). It is thoughtthat instabilities in the mixed layer may serve as afundamental link between mesoscales and small scales(Boccaletti et al. 2007, Fox-Kemper et al. 2008), andcould be important in order to construct the full pictureof oceanic multi-scale turbulent interactions (McWil-liams 2008). The generation of submesoscale flows is aninteresting problem because the behaviour of flows in

geostrophic balance in the high-aspect ratio of oceandomains tends to be analogous to two-dimensional (2D)turbulence, where the energy cascade is primarilytoward larger scales, namely away from submesoscales.Second, vertical biogeochemical transport due tosubmesoscale filaments created by intermittent turbu-lent events can be as much as those associated withmesoscale transport events in the ocean (Mahadevanand Tandon 2006, Klein and Lapeyre 2009). Third,these processes are of significant importance for navaloperations, navigation, submerged equipment such assubmarines, acoustic propagation and transport ofmines. For these reasons, the US Office of NavalResearch (ONR) formed a team of investigators tomodel and observe upper ocean lateral mixing pro-cesses. These processes are likely to include mixed layerinstabilities, which have not been observed before (tothe knowledge of the authors). One of the ONR fieldexperiments involves three ships, an airplane and manyobservational platforms, and it is one the mostcomprehensive observational cruises in recent times.

In this manuscript, we present a short overview ofmodelling and observational approaches for submesos-cale ocean flows, as well as a computationally demand-ing mixed layer dynamics study using a CFD code,subject to sampling constraints of ocean experiments.

1.2. Considerations on modelling approaches forsubmesoscale ocean flows

Traditional ocean general circulation models(OGCMs) have the advantage that they are already

*Corresponding author. Email: tozgokmen@rsmas.miami.edu

International Journal of Computational Fluid Dynamics

2012, 1–12, iFirst article

ISSN 1061-8562 print/ISSN 1029-0257 online

� 2012 Taylor & Francis

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configured at the global scale, as well as for specificocean basins by national operational centers, such asNaval Research Laboratory (NRL) and NationalCenter for Environmental Prediction (NCEP)1, withmesh spacings going down to about 5 km. OGCMscan also be set up for regional simulations at finerresolutions (Capet et al. 2008). The primary advantageof these approaches is that the dynamics of the generalcirculation and mesoscale features such as jets andeddies are well captured. These OGCMs containrealistic forcing, domain geometry, and assimilateocean data.

There are however several disadvantages ofOGCMs for simulating submesoscale flows. First, theequation set (so-called primitive equations (PE))contains the hydrostatic approximation, which isjustified by the high aspect ratio between horizontaland vertical domain dimensions. The hydrostaticapproximation becomes invalid for scales belowapproximately 1 km (Kantha and Clayson 2000),thus within the submesoscale range. The breakdownof the hydrostatic approximation creates two types oferrors in OGCMs. Internal waves are prominentfeatures in the ocean with well-known dispersionrelations and characteristics (Garrett and Munk1972). It is known that hydrostatic models cannotproduce the correct dispersion for non-linear internalwaves, leading to unrealistically fast propagatingwaves (Scotti and Mitran 2008). Also, the overturningof density surfaces by Kelvin-Helmholtz instabilities,which are one of the primary mechanisms responsiblefor mixing in the ocean, cannot be explicitly capturedwith hydrostatic models. In fact, the accuracy ofhydrostatic models can substantially degrade with finerresolution at such scales (Chang et al. 2005). Inclusionof a non-hydrostatic pressure solver in hydrostaticOGCMs requires a substantial change in these codes(Scotti and Mitran 2008).

Second, for the reasons described above, OGCMscontain parameterisations for mixing and dissipation(but not for dispersion, to the knowledge of theauthors). These parameterisations may range fromalgebraic models to second-order turbulence closures(Large 1998). One challenge with algebraic closures isthat they can contain dimensional parameters thatneed to be tuned for different flow problems (Changet al. 2005). The turbulence closures have beenimported from the engineering community and furtherdeveloped for stratified oceanic flows (Mellor andYamada 1982, Kantha and Clayson 1994, Burchardand Baumert 1995, Burchard and Bolding 2001,Canuto et al. 2001, Baumert and Peters 2004, Baumertet al. 2005, Umlauf and Burchard 2005, Warner et al.2005, Canuto et al. 2007). They have been shown towork well in challenging ocean mixing problems (e.g.

Ilicak et al. 2008). But perhaps they require moreextensive evaluation as more comprehensive oceandata become available. Also, given that the hydrostaticapproximation changes the vertical momentum bal-ance, turbulence closures are applied only in thevertical direction. This has created a disconnectbetween horizontal and vertical closure schemes.

Third, there are challenges with data assimilation.Assimilation can influence conservation laws. Inaddition, it is not clear that assimilating data thatcontain much higher spatial resolution than theOGCMs, or simply point measurements on 5–10 kmgrids would enhance the realism of the model forsubmesoscale processes. Finally, the prevailingOGCMs are based on second order numerics, therebyrequiring more mesh points for convergence thanhigher order methods (Deville et al. 2002). Given thatsubmesoscale processes are expected to be capturednear the highest resolved wave number range inOGCMs, it becomes likely that numerical dissipationand dispersion errors could influence submesoscaleflow behaviour.

As such, it is beneficial to consider methodsdeveloped in the CFD community for simulatingsubmesoscale flows. While these models are typicallynot configured to incorporate the large-scale realism ofOGCMs, they offer high numerical accuracy, scal-ability, adaptive and/or unstructured meshes, andintegrate the well-established Navier–Stokes or Bous-sinesq equations (BE) of motion. In that sense, theyaddress many of the issues that create challenges forOGCMs outlined above. Nevertheless, not all CFDmethods are appropriate for oceanic problems. Thelarge Reynolds numbers in ocean flows prohibit thedirect numerical simulation (DNS), in which alldegrees of freedom are computed. Reynolds averagedNavier Stokes (RANS, Wilcox 1998) codes arecomputationally efficient, but raise several questions.First, the decomposition of the flow field into a meanand fluctuating component creates some ambiguity fortransient flows, as in the case of mixed layer instability.Second, many RANS closures rely on the eddyviscosity paradigm, or forward cascade of energy,which is not always valid in the presence of stratifica-tion, and, in particular, rotation (Vallis 2006). Analternate approach is the large eddy simulation (LES,Ferziger 2005), which lies in between the extremes ofDNS, in which all turbulence is resolved, and RANS,in which all turbulence is modelled. In regards to thenumerical method, the spectral element method(SEM), which combines the geometrical flexibility offinite element method (FEM) with the numericalaccuracy of spectral models (Patera 1984, Maday andPatera 1989), seems to be ideally suited for subme-soscale oceanic flows.

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Here, we employ Nek5000, which is a mature DNSand LES model based on the SEM developed by P.F.Fischer2 and colleagues. It has been used in oceanrelated research (Ozgokmen et al. 2004a,b, 2006,Chang et al. 2005, Xu et al. 2006, Ozgokmen et al.2007, Ozgokmen and Fischer 2008, Ilicak et al. 2009;Ozgokmen et al. 2009a,b) and most recently in aninvestigation of mixed layer instability (Ozgokmenet al. 2011).

1.3. Considerations on observational approaches forsubmesoscale ocean flows

Submesoscale flows would ideally require 4D (x,y,z,t)sampling of velocity and density fields. But, the largevolume of water makes representative measurements adifficult, or even a daunting challenge (Sanford et al.2011). Traditional instruments such as current metermoorings collect fixed point measurements of horizon-tal velocity components, and are not well suited forflows with a high degree of spatial structure. Shipboardinstruments for measuring density fields (such asconductivity temperature depth (CTD)) and velocity(such as acoustic Doppler current profiler (ADCP))allow flexibility in spatial sampling, but they are slavedto the location of the ship, thereby requiring multipleships for simultaneous surveys. Vertically profilingautonomous instruments such as gliders (Webb et al.2001) offer a cost-efficient alternative to ships for long-term monitoring of water properties. But profilinginstruments create a compromise between vertical andhorizontal coverage. Current meters installed onautonomous vehicles measure only the relative velocityof the device through the surrounding water. Also,rapidly moving internal wave displacements createaliasing errors in glider profiles, since each of thevertical cycles can take many hours to complete.

There are two remote sensing approaches that offerhorizontal snapshots of ocean fields. The first issatellite or aircraft based sensors. Altimeters detectsea surface height anomaly, from which an estimate ofocean currents can be obtained through the assump-tion of geostrophic momentum balance. Nevertheless,current satellite products3 offer only mesoscale-resol-ving capability. Even when higher resolution productsbecome available, errors associated with the Earth’sgravity anomalies and waves can become comparableto the ocean’s surface topography signal created bysubmesocale flows. Ocean colour sensors4 revealsignificantly more detail at much high resolutions.Nevertheless, they do not detect any of the primaryvariables of the equations of motion, but ratherchlorophyll, which is mostly limited to coastal regionsand represents a depth-averaged distribution of aquasi-passive scalar field.

High frequency (HF) radar is an exciting tool withspatial and temporal resolutions down to about 250 mand 15 min, respectively, providing true snapshots ofsurface velocity fields over length scales of 10–100 km(Steward and Joy 1974, Paduan and Rosenfeld 1996,Kaplan et al. 2005, Shay et al. 2007). In fact,submesoscale features have been detected by an HFradar (Shay et al. 2003). These instruments arebecoming widely used in coastal experiments (Hazaet al. 2010). Nevertheless, HF radars are limited tocoastal zones and surface flows at the present moment.

There are two observational techniques that arequite well suited for sampling submesoscale flows. Thefirst is the use of Lagrangian particles. While Lagran-gian drifters and floats have been used in oceanogra-phy for a long time to detect ocean currents (Davis1985, Lumpkin and Pazos 2007), it is only recently thatsimultaneous launches of drifter clusters are being usedas a technique to infer measures of multi-scale stirringin the underlying fluid (Koszalka et al. 2009, Lumpkinand Elipot 2010, Schroeder et al. 2011). The second isupper ocean dye release that can be detected usingairborne laser (Sundermeyer et al. 2007), or LIDAR.Since the LIDAR is able to sample with approximately200 m swath from an airplane flying with 100 m/s, it iscapable of providing truly synoptic horizontal fields ofupper ocean tracer concentration. Both platforms areessentially Lagrangian sampling methods that avoidmany of the aliasing errors of some other approaches.

1.4. Objectives

Our main goal here is to create fields of mixed layerinstability using a CFD code, and then examine theeffectiveness of passive tracers and drifters in samplingthese submesoscale flows. This study is a continuationof a recent effort (Ozgokmen et al. 2011), but differs inthat we consider sampling strategies subject to thefollowing experimental constraints:

(a) The flows of interest take place inside mixedlayers that are about 25 m deep, when com-pared to the lateral size of interest of 104 m.Modelling of flows with such high geometricaspect ratio of 1/400 demands schemes withlow numerical dissipation and the use of highPeclet numbers in order to avoid an unrealisticconversion of available potential energy drivingthe system to background potential energy byspurious mixing, instead of kinetic energy viaflow instabilities.

(b) The tracer patches are likely to be approxi-mately 500 m long, 50 m wide and 2 m thick(all dimensions are estimates within a factor ofabout four). The tracer is to be injected at the

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mixed layer base. This is mainly to reduce theeffects of the diurnal cycle and of wind-drivenflows near the surface on tracer evolution.

(c) Due to the decay rate of fluorescein needed forairborne detection of the dye by laser, theLIDAR (Sundermeyer et al. 2007) observationperiod is limited to 36 h, while other dyesampling can last up to 120 h.

(d) About 18 drifters are available with a temporalsampling of 30 min for a total period of about30 days.

These constraints are by no means the only onesthat will be encountered in the field. Here, we willassume that there are no spatial or temporal errors inobserving the dye or drifters, and that the velocity fieldis frozen during the period of launching the observa-tional assets. Surface or drogued drifters are subject toposition errors of about 10 m (or more) via the globalpositioning system as well as slippage (Ohlmann et al.2005). As such, the assumption of no spatial andtemporal errors in drifter observations is a strong oneand this topic can benefit from investigation in the nearfuture.

2. Model setup and parameters

Nek5000 is configured to solve the following equationset:

Du

Dt¼ Ro�1H z� u�r�p� Fr�2 r0 zþ Re�1r2u�r � t ;

ð1Þ

r � u ¼ 0 ; ð2Þ

Dr0

Dt¼ Pe�1r2r0 � r � s ; ð3Þ

DC

Dt¼ Pe�1r2C ; ð4Þ

dx

dt¼ u ; ð5Þ

where the variables are the velocity field u, pressure p,density perturbation r0, passive scalar field C andpassive particle positions x. The non-dimensionalparameters are the Reynolds number Re¼U0H/v, thePeclet number Pe¼U0H/k, the Froudenumber Fr¼U0/(N/H), and the vertical Rossbynumber RoH¼U0/(fH)¼ aRo, where Ro¼U0/(fL) isthe Rossby number and a¼L/H the ratio of horizontaland vertical domain sizes. U0 is the flow speed scale, vis the kinematic viscosity, k is the molecular diffusiv-ity, g is the gravitational acceleration, r0 is the fluid

density, N is the buoyancy frequency, f is the Coriolisfrequency, z is the unit vector in the vertical direction.Not all scales of motion are resolved, but consistentwith LES formalism, it is assumed that the restrictionof the computed fields to the numerical grid constitutesthe convolution procedure leading to spatially filteredvariables u, r0 and �p. A dynamic Smagorinsky modelis used for the subgrid-scale stress tensort ¼ �u�u� �u�u ¼ t ¼ �2 ðcdsdÞ2 jrsuj rsu ;where rsu :¼ðruþruTÞ=2 is the deformation tensor of u, and cds iscomputed using the dynamic procedure (Germanoet al. 1991). The subgrid-scale vector is taken ass ¼ ur0 � ur0 ¼ 0, following fairly extensive tests ofstratified mixing by Ozgokmen et al. (2009a,b).

The domain is a L6L6H box in x, y, zdirections, respectively, with L¼ 104 m, and H ¼500 m. Periodic boundary conditions are applied inthe x direction, while free-slip, insulated walls areplaced in the other directions. The initial condition isgiven by the analytical function

r0ðx; 0Þ ¼ 1� 0:7z

H

� �2:3� �

1� exp � y=L

0:5þ E

� �4

� 20

�1� z

H

�� �8( )" #

; ð6Þ

which generates a h0¼ 25 m deep mixed layer andseveral km wide front perturbed by E ¼0:08cos ð2p x

LÞ þ 0:03cos ð5p xLÞþ0:005cos ð20p

xLÞþ 0:01

cos ð30p xLÞ (Figure 1a). The depth of the mixed layer is

chosen to reflect h0 in the Sargasso Sea in the summer,corresponding to the season of one of the ONRcruises. The turbulent exchange across the front, asquantified by an eddy diffusivity coefficient, is propor-tional to h20 (Fox-Kemper et al. 2008). The wide rangeof wave lengths used in the perturbations reduce thespin-up time of the computation by letting the fast-growing modes emerge efficiently. The simulation isstarted from rest and contains no external (e.g. wind)forcing.

In selecting the parameters of the system, it is notedthat values as high as Re¼ 105 and Pe¼ 76 105 (andhigh resolution) are needed in order control verticaldiffusion across the thin mixed layer. The otherparameters are set to Fr¼ 0.1 and Ro¼ 0.02, as inOzgokmen et al. (2011). The fastest growing modeR inmixed layer instability is approximated by Rd=R � 0:2

(Eldevik and Dysthe 2002), where Rd ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigr0Dr0m h0

q=f

is the radius of deformation in the mixed layer withDr0m being the density change across the surface front.

The number of largest coherent features that can fit in

the domain is estimated from LR � 0:2 Fr

RoDr0mDr0

h0H

� ��1=2.

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Substituting Dr0m/Dr0 � 0.2 and h0/H¼ 25/500, we get

L=R � 10.The domain is discretised using Kx¼Ky¼ 32 and

Kz¼ 8 elements in the horizontal and vertical direc-tions, respectively. The horizontal elements have uni-form spacing while those in the vertical are clusterednear the mixed layer. On each element, solutions arerepresented by Legendre polynomials of order N ¼ 11,leading to the total number of grid points of n ¼ðKxN þ 1ÞðKyN þ 1ÞðKzN þ 1Þ ¼ 11; 090; 201. TheGauss–Labatto–Legendre quadrature points have aminimum and maximum spacings of ðDxÞmin � ‘ 5

N 2

and ðDxÞmax � ‘ p2N for a given element length ‘,

resulting in 9 m � Dx �43 m and 0.4 m � Dz �1.8 min the mixed layer. The computations are carried outon 128 processors using Virginia Tech’s SystemX, aswell as on IBM-P5 and Linux clusters at the Universityof Miami. The time step is equal to 150 s, for which theCourant number stays less than 0.4. The system is

integrated for a total of about 29.5 days (about 17,000steps), which takes a wall clock time of 24 h.

3. Results

3.1. Description of the flow field

From the initial perturbation e consisting of variouswavelengths, modes consistent with the scale ofR growfastest and towards the higher r0 (northern side) of thefront during the first week of the instability (Figure1a,b). The circulation inside these modes is anticyclonic(clockwise or negative relative vorticity, z). Under theassumption that tilting/twisting and solenoidal terms inthe vorticity budget are negligible, the potential vorticity(PV) can be expressed as q¼ (fþ z)/h, so that the initialPV in the mixed layer is q1¼ f/h0. During the rotatinggravitational adjustment, the water parcels in the mixedlayer are advected such that their height h1 reduces.Under the PV conservation q1¼ f/h0 (fþ z1)/h. Since

Figure 1. Perspective view of the density perturbation field r0 at (a) t¼ 0, (b) t¼ 11 days, (c) t¼ 17.3 days, and (d) t¼ 29.5 days.The 3D plot is sliced in order to reveal the isopycnal at the base of the mixed layer. The surface plot shows the r0 ¼ 0.22 isopycnal,marking the base of the mixed layer. The vertical axis is stretched 10 fold with respect to the realistic aspect ratio. The animationis available from: http://www.rsmas.miami.edu/personal/tamay/3D/mli116-tr.mov.

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h15 h0, z15 0, or anticyclonic circulationmust developinside these features. The zones in between these modesare subject to shear of reverse direction (cyclonic), whichis also supported by PV conservation arguments for thedeeper layer; q2 ¼ f=ðH� h0Þ ¼ ðfþ z2Þ=h2. Since h25(H7h0), one gets z24 0. We note that the shear layersbetween the fastest growing surface modes continue torelease small cyclonic vortices, that populate the mixedlayer base (Figures 1b and 3b). There are occasionalindications of Kelvin-Helmholtz type shear instabilitybetween the surface modes (Figure 1c). The weaker yetpersistent geostrophic current caused by the front (fromleft to right of the domain, or with the deeper, highpressure perturbation pool on its right) appears toinfluence the flow evolution. First, it tends to disrupt thefastest growing modes, causing detachment of antic-yclonic surface eddies. At longer times, anticyclonicsurface eddies spin down, and the cyclonic shear alongthe northern side of the front creates a large number(about 10) of cyclonic spiraling eddies near the surface.Spiral eddies are commonly observed in the ocean, andtheir formation in this simulations appears to beconsistent with the mechanism outlined in Figure 34 ofMunk et al. (2000).

3.2. Sampling with passive scalars

Realistic size tracer strips are used to sample themixed layer base. One of the striking first impressionsis how small the dye patches are with respect to thesubmesoscale features in the flow field (Figure 2).Subsequently, multiple patches need to be considered,and positioning of the patches could play animportant role given the limited observation period.It is assumed that the topology of the mixed layerbase can be mapped to some degree of accuracy usingship board and/or profiling instruments. Two types offeatures are selected. The first type of features arecenters of eddies that populate the mixed layer base(Figure 2a). These are essentially elliptic stagnationpoints that tend to trap particles. The second type offeatures are regions in between the eddies that arehyperbolic due to strain exerted by the eddies (Figure2b). In order to accelerate the formation of tracerpatterns, 10 patches of tracers are launched in a crossconfiguration (five independent locations) at themixed layer base at 25 m. The launch time is chosento be t¼ 17.3 days (Figure 1c), at which point thesystem exhibits well-developed instabilities, whileallowing adequate time for integration before theclosed boundaries influence the turbulent features.

Figure 2c,d indicates that rotation and trappingassociated with the elliptic eddies, and stretching alongthe out-flowing manifolds of hyperbolic regions inbetween the eddies are apparent in the tracer fields

sampled after 36 h. This result shows the feasibility ofidentifying some of the pronounced features in themixed layer base using well-resolved sampling of dye,despite the translation of features by several km overthe observation period.

Nevertheless, one of the primary obstacles asso-ciated with dye measurements is that the concentrationfield C is not easy to convert into useful diagnostics oftransport. For instance, while the dye behaviour inFigure 2 is qualitatively indicative of elliptic andhyperbolic regimes in the underlying flow field, we arenot aware of any quantitative methods of how toconvert C into eigenvalues of the velocity gradienttensor, or compute components of strain tensor. This ismainly due to two reasons. First, dye parcels cannot betagged (apart from using drifters), and second,formation of gradients in the dye field by the actionof coherent features starts to become compensated bydiffusion in Equation (4) at some point, most notablyalong the exponentially stretching (hyperbolic)branches of the flow field. It is common to computesecond moment, or spreading around the center ofmass from tracer measurements (Sundermeyer andLedwell 2001). But it is not clear to us how to handlethis in the spreading is highly anisotropic and only asmall fraction of the turbulent features are sampled asin Figure 2. Comparison of tracer and particlehorizontal position variances and extension to verticaldiffusivity was presented in Figures 19 and 23 ofOzgokmen et al. (2011). We have computed thefollowing metric, w2ðtÞ ¼

RjrCðx; tÞj2 dV=

RjC

ðx; tÞj2 dV, introduced by Pattanayak (2001) andSundaram et al. (2009) to quantify this competitionbetween chaotic advection and diffusion, but found nosignificant difference between the elliptic and hyper-bolic launches during the anticipated 36 h, and evenafter the maximum 120 h of observation period.

3.3. Sampling with passive particles

The main disadvantage of drifters with respect to tracersis that they provide sparse information, as it is notfeasible to release thousands of them in ocean experi-ments, given present technology. Nevertheless, particlesoffer two advantages over tracers. They are not subjectto diffusion, and they provide position information,from which it is quite easy to compute quantitiesrelevant to transport. A diagnostic of interest is thescale-dependent Lyapunov exponent (Finite Scale Lya-punov Exponents (FSLE), Artale et al. 1997, Aurellet al. 1997), which is directly connected to turbulent andscalar fluctuations in the underlying flow field:

lðdÞ ¼ lnðaÞhtðdÞi ; ð7Þ

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where ht(d)i is the averaged time (over the number ofparticle-pairs) required to separate from a distanceof d to ad. The FSLE has been used in many recentstudies of oceanic dispersion (Lacorata et al. 2001,LaCasce and Ohlmann 2003, Haza et al. 2008,Koszalka et al. 2009, Haza et al. 2010, Lumpkin andElipot 2010, Poje et al. 2010, Schroeder et al. 2011).

In order to obtain an accurate evaluation of l(d) inthis flow simulation, the field is sampled over an areaof 2 km6 5.5 km using 3528 particles launched astriplets at 25 m depth (Figure 3a). The particlescomposing the triplets are released 20 m apart, andthe triplets are 100 m apart. Particles are advectedonline for about 30 days with the model time step of150 s, while the positions are recorded at 30 minintervals, as per observational requirements.

The resulting l(d) plot (Figure 4) identifies aconstant-value plateau for the scale separation rangefrom 20 m � d � 600 m and a fall off to Richardsonregime for larger separation distances. The fall-offscale is related to the average eddy diameter in thesampled field, while the value of the maximum FSLE isthe average stretching that the particles experience(Poje et al. 2010). The maximum value seems to beabout lmax � 0.4 days71. Estimates from the observa-tional and modelling studies listed above are in therange of 0:3 � lmax � 10 days71. The present value ison the lower end of the spectrum of estimates,primarily because all others are for near-surface flows.A previous study on mixed layer LES (Ozgokmen et al.2011) revealed a maximum Lyapunov exponent lmax�2 days71, with the primary differences being that the

Figure 2. Top view of the domain showing the initial locations of realistic size tracer strips in cross configuration positioned tosample (a) eddies in the mixed layer base, and (b) regions in between the eddies. The surface plot shows the r0 ¼ 0.22 isopycnal,marking the base of the mixed layer, at t¼ 17.3 days. The vertical axis is stretched 30 fold to emphasise the turbulent coherentfeatures. The state of the tracer strips (contoured using 0.1 within non-dimensional range of 0 � C �1) after 36 h are depicted in(c) and (d). The animations are available from: http://www.rsmas.miami.edu/personal/tamay/3D/mli121-tr-v1.mov and http://www.rsmas.miami.edu/personal/tamay/3D/mli122-tr-v1.mov.

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instability of a thicker (h0¼ 80 m) mixed layer wassampled at 5 m depth.

Then, the following question is posed:

. How do we target a much smaller number ofparticles (18) to approximate this l(d) curve withthe least error?

It is helpful to use a quantitative criterion to select

the target areas. Here we employ the Okubo–Weiss

(OW) criterion (Okubo 1970, Weiss 1991), Q ¼S2 � o2 ¼

@u@xþ

@v@y

2 þ 4@u@y

@v@x�

@u@x

@v@y

, where S2 ¼

Figure 3. (a) Okubo–Weiss (OW) parameter, normalised by midlatitude Coriolis parameter (f¼ 76 1075 s71), Q0/f 2, at 25 mdepth and t¼ 17.3 days, as well as initial positions of 3528 particles launched in triplets. (b) Targeted launches of triplets in sixstrain dominated regions (Q04 0, marked with ‘1’), in six vorticity dominated regions (Q05 0, marked with ‘2’) and six regions oflow Q0 (marked with ‘3’).

@u@x�

@v@y

2 þ @v@xþ @u@y

2is the square of the horizontal

strain rate, o2 ¼@v@x�

@u@yÞ

2 the square of horizontal

vorticity. The OW criterion is a useful Eulerian metricto differentiate the hyperbolic (Q4 0) and elliptic(Q5 0) parts of a two-dimensional flow field. It isassumed that data from other in situ instruments canhelp construct an approximate OW map to help guidetargeted drifter launches. The construction of an OWmap is a challenging task using ship-board instru-ments, since this requires a synoptic view of thehorizontal velocity field. But it is not impossible if twoor more ships follow parallel tracks with a good sense

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of aliasing errors, or if several dozens of profiling floatsare employed. Contamination of the signal associatedwith the coherent features by the internal wave fieldcould also be another issue that is likely to beencountered in the field. For further simplicity, onlythe non-divergent part of the OW is used,Q0 ¼ 4

@u@y

@v@x�

@u@x

@v@y

. Scaling with a midlatitude Cor-

iolis frequency shows that extrema of jQ0j/f 2 are on theorder of 1 (Figure 3a), indicative of high strain andvorticity values characteristic of submesoscale flows(Thomas et al. 2008).

A set of five targeted releases are carried out(Figure 3b). The first release contains six triplets atstrain-dominated regions (Q04 0, between eddies), thesecond in elliptic regions (Q04 0, inside eddies), andthe third in Q0 � 0 regions. The fourth and fifthlaunches contain two realisations of random combina-tions consisting of two of each type of release (so-called mixed releases).

Figure 4 shows that the first launch starts with highFSLE (as expected) but drops almost to the true curve(from 3528 particles) at d � 100 mm, which seems tobe the scale that the particles remain inside high straininitial regions. The launches in region two show lowFSLE up to d � 600 mm (the average size of theeddies, as expected). The mixed launches are possibly

the best performing. Nevertheless, perhaps the mainmessage is that almost all of the releases are leading toquite accurate results, because there are many eddiesand high hyperbolicity in the mixed layer base, andgiven enough time, 18 particles seem to sample thedispersive statistics of these features quitesatisfactorily.

While particle/tracer variances are accurate mea-sures of how the turbulent exchange widens an oceanfront, they may not be necessarily the best measures ofsubmesoscale features which tend to lead to rapidlystretching filaments, in a significant deviation from adiffusive processes. The scale-dependent FSLE pre-sented in the manuscript is an accurate statisticalmeasure of the strain rate (lmax) induced by sub-mesoscale eddies, and their scale (approximate widthof the FSLE plateau). As Figure 4 indicates, theregimes of the l(d) are super diffusive at all separationscales.

4. Summary and future directions

The primary original aspects of this study and ourmain findings are highlighted below. Several commentson the future of CFD-based approaches in oceanmodelling follow.

Figure 4. The scale-dependent FSLE l(d) from the full set with 3528 particles, and targeted sets each with 18 particles. Theaverage FSLE from five launches with 18 particles is also shown. The slope in the background indicates the Richardsondispersion regime (l* d72/3).

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(a) To our knowledge, this is the first numericalstudy in which such shallow mixed layerinstabilities have been explicitly resolved, pri-marily due to lack of numerical dissipation ofthe high-order numerical model. In addition,the high resolution of the model allows thesimulation of realistic size tracer patches. Bothaspects highlight some of the capabilities ofCFD-based ocean modelling approaches overmore conventional, parameterisation-basedavenues.

(b) Our results put the use of realistic-size tracersin perspective. Larger amounts and/or longerobserving periods than those used here seem tobe preferable to sample the eddies at the mixedlayer base. Alternatively, if ocean experimentsindicate much larger rates of tracer spreadingthan those observed here, then this wouldimply the importance of missing processes inour idealised configuration.

(c) We conclude that the results presented here aresupportive of the effectiveness of Lagrangiandrifters in sampling rapidly evolving subme-soscale mixed layer flows. Future availability ofcheaper, smaller and environmentally degrad-able drifters may permit their deployment inlarge numbers [on the O(100)] for betterobserving capabilities.

(d) To our knowledge, this is one of the fewcomputations in which spiral eddies, oftenobserved in the ocean (Munk et al. 2000),emerge spontaneously. We provide support tothe idea put forward by Eldevik and Dysthe(2002) that spiral eddies originate from frontalinstabilities at the ocean’s mixed layer.

(e) We also provide a brief overview of how CFD-based approaches can contribute to oceanmodelling and in situ sampling.

Presently, LES and OGCMs are fully complemen-tary in that LES models provide more reliable results forscales smaller than the submesoscales, while OGCMscontain realistic mesoscale currents and general circula-tion. Both modelling approaches overlap over thesubmesoscale range. It is not a practical long-termsolution to try to couple an LES model to OGCMs dueto large differences in their architecture. The disconti-nuity between BE and PE sets is most notable whenconsidering such multi-scale simulations. Assuming asteady growth in computing power, we envision a hardbarrier for OGCMs at about the submesoscale range,while LES models can continue to expand their frontiertowards larger scales. Certainly, adaptive mesh refine-ment (AMR) methods within FEM/SEM can be usedfor multi-scale simulations (Mavriplis 1994). A hybrid

approach is to solve BE over a finer mesh andcomputationally cheaper PE over coarser mesh regionsof the domain within the same code (Botelho et al.2009). Nevertheless, assuming that about half of themesh points are dedicated to coarse grid parts of thedomain, a speed up factor of at most two does notencourage dealing with potentially complex dynamicalmismatches arising from different equation sets in thesame domain. A third idea is to find intermediateequation sets which smoothly bridge the BE and PE(Duan et al. 2010). But, such approaches have not beentested yet. Therefore, a reasonable avenue for a multi-scale ocean modelling system is to rely only on LEScodes, like Nek5000, for which we do not see any majorcomplications regarding future development. Perhapsone aspect that will gain increasingly more attention isthe post-analysis of large outputs that will be inevitablygenerated by multi-scale computations. It will notalways be possible to estimate a priori which measuresare of interest for a specific problem. Running thescalable main code on a large number of cores multipletimes is likely to be more efficient for generating on-linediagnostic metrics than using off-line post-processingtools on a smaller number of processors.

Acknowledgements

We greatly appreciate the support of the Office of NavalResearch and National Science Foundation via grants N00014-09-1-0267, DMS-1025323, DMS-1025359 under the Colla-borative Mathematics and Geoscience (CMG) initiative. Thisresearch was made possible in part by a grant from BP/TheGulf of Mexico Research Initiative. We thank all members ofthe Lateral Mixing DRI team, who have provided guidancethrough many insightful comments. The computations werecarried out on the University of Miami’s high-performancecomputing (HPC) center (http://ccs.miami.edu/hpc/), on Sys-temX at Virginia Tech’s advanced research computing center(http://www.arc.vt.edu) and at the City University of NewYork High Performance Computing Center under NSF GrantsCNS-0855217 and CNS-0958379. We thank the four anon-ymous reviewers whose comments resulted in a significantimprovement of the manuscript. We greatly appreciate theleadership of Catherine Mavriplis in organising this timelyspecial issue, and many useful comments and guidance sheprovided during the preparation of this manuscript.

Notes

1. http://www.hycom.org/ocean-prediction2. http://nek5000.mcs.anl.gov/index.php/Main_Page3. http://www.aviso.oceanobs.com/es/data/products/index.

html4. http://oceancolor.gsfc.nasa.gov/cgi/image_archive.cgi?c¼

CHLOROPHYLL

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