Lagrangian predictability characteristics of an Ocean Model

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1002/,

Lagrangian predictability characteristics of an Ocean1

Model2

Guglielmo Lacorata, Luigi Palatella, Rosalia Santoleri

G. Lacorata, Istituto di Scienze dell’Atmosfera e del Clima, Consiglio Nazionale delle Ricerche,

Via Monteroni, I-73100, Lecce, Italy. ([email protected])

L. Palatella, Istituto di Scienze dell’Atmosfera e del Clima, Consiglio Nazionale delle

Ricerche, Via Monteroni, I-73100, Lecce, Italy and INFN sez. Lecce ([email protected];

[email protected])

R. Santoleri, Istituto di Scienze dell’Atmosfera e del Clima, Consiglio Nazionale delle Ricerche,

Via Fosso del Cavaliere, I-00133, Roma, Italy. ([email protected])

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Abstract.3

The Mediterranean Forecasting System (MFS) Ocean Model, provided by4

INGV, has been chosen as case study to analyze Lagrangian trajectory pre-5

dictability by means of a dynamical systems approach. To this regard, nu-6

merical trajectories are tested against a large amount of Mediterranean drifter7

data, used as sample of the actual tracer dynamics across the sea. The sep-8

aration rate of a trajectory pair is measured by computing the Finite-Scale9

Lyapunov Exponent (FSLE) of first and second kind. An additional kine-10

matic Lagrangian model (KLM), suitably treated to avoid “sweeping”-related11

problems, has been nested into the MFS in order to recover, in a statistical12

sense, the velocity field contributions to pair particle dispersion, at mesoscale13

level, smoothed out by finite resolution effects. Some of the results emerg-14

ing from this work are: a) drifter pair dispersion displays Richardson’s tur-15

bulent diffusion inside the [10-100] km range, while numerical simulations16

of MFS alone (i.e. without subgrid model) indicate exponential separation;17

b) adding the subgrid model, model pair dispersion gets very close to observed18

data, indicating that KLM is effective in filling the energy “mesoscale gap”19

present in MFS velocity fields; c) there exists a threshold size beyond which20

pair dispersion becomes weakly sensitive to the difference between model and21

“real” dynamics; d) the whole methodology here presented can be used to22

quantify model errors and validate numerical current fields, as far as fore-23

casts of Lagrangian dispersion are concerned.24

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1. Introduction

The capability of predicting the time evolution of a physical system, given its present25

state, stands at the very foundations of scientific knowledge, with obvious applications to26

a wide range of issues, like, for instance, the study of weather forecasts, climate changes,27

population dynamics, tracer dispersion, etc. Once a specific model of a given physical28

system has been written, the state of the system can be, in principle, computed at any29

time by numerically integrating the evolution equations. The major problem arising when30

doing numerical simulations on a complex non linear system is the growth of perturbations,31

or errors, along the time evolution of the state [Lorenz , 2006]. The error growth is32

substantially due to:33

i) strong sensitivity to initial conditions;34

ii) uncertainty affecting the physical parameters appearing in the model, as well as the35

simplified form of the model equations with respect to the actual dynamics.36

In this work we will refer to case i) as first-kind predictability problem (PP-I), and to37

case ii) as second-kind predictability problem (PP-II). Studies on PP-I are far more nu-38

merous in literature, since the phenomenon of chaos in dynamical systems theory has been39

theoretically, and methodologically, well formalized [Ott , 2002]; PP-II, on the contrary,40

still lacks a systematic approach, even though some attempts in that direction have been41

made by some authors [Boffetta et al., 2000a; Iudicone et al., 2002].42

In chaotic systems, the leading Lyapunov exponent, λ1, determines a time scale of43

predictability, TP ∼ 1/λ1, as long as the error on the state can be considered infinitesimal.44

A more accurate analysis of the problem consists in measuring trajectory pair separation45

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on finite scales, and thus relating the predictability time to a given scale of motion. This is46

accomplished by measuring the Finite-Scale Lyapunov Exponent λ(δ), or FSLE [Aurell et47

al., 1996, 1997], i.e. the mean growth rate of a perturbation as function of the perturbation48

size δ. Clearly, the FSLE has a major relevance in all cases where one has to deal with49

realistic perturbations of significant size, e.g. Lagrangian tracer dispersion on scales of50

the same order as the size of the coherent structures of a marine or atmospheric system.51

When applied to case i), i.e. for trajectory pairs evolving in the same dynamical system,52

the FSLE will be called first-kind FSLE, or FSLE-I; when applied to case ii), i.e. when53

one trajectory of a pair belongs to the model and the other one to the “real” system, it54

will be called second-kind FSLE, or FSLE-II [Boffetta et al., 2000a].55

In numerical Lagrangian studies, e.g. simulations of tracer dispersion in ocean or atmo-56

sphere, one usually deals with a given velocity field from which passive tracer trajectories57

(fluid particles) can be computed by time integration. Formally, we can circumscribe the58

problem to the case:59

dx

dt(t) = U(k0,ω0)(x, t) + u(k0,ω0)(x, t) (1)60

In (1), we define: x as the n−dimensional position vector of a tracer particle; U as the61

“large-scale” velocity field at wavenumbers k ≤ k0 and frequencies ω ≤ ω0; u as the “small-62

scale” velocity field at wavenumbers k > k0 and frequencies ω > ω0; l0 ∼ k−10 and t0 ∼ ω−1

063

as space and time resolution scales of the dynamics, respectively. In the following, we will64

refer to system (1) as the “real” Lagrangian dynamics, in the hypothesis that both U and65

u fields are known with infinite precision, i.e. if no component of the dynamics is missing.66

Consequently, Lagrangian trajectories x(t) from Eq. (1) will be considered as the “true”67

tracer particle trajectories. In all realistic applications, we deal with model trajectories,68

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y(t) = x(t), generated by the resolved field (U), while the unresolved dynamics (u) is69

either neglected or, possibly, replaced by some suitable parametrization:70

dy

dt(t) = U(k0,ω0)(y, t) + u(y, t) (2)71

Even in case of “ideal model”, small errors on the initial conditions typically will tend72

to grow exponentially in time, because of the non linearity always present in all realistic73

physical systems. Normally, the exact dynamics is unknown, so that also the differences74

existing between the “real” equations of motion, Eq. (1), and their approximated expres-75

sions, Eq. (2), play a relevant role in limiting the predictability of the trajectories.76

From a very applicative point of view, one would like to exploit Lagrangian studies in or-77

der to gather useful information for implementing or improving numerical tools that could78

allow one to model, with a given accuracy level, physical observables such as temperature79

and salinity, chemical properties, pollutants, floating debris, particulate and sediments as80

well as biological tracers such as phytoplankton, zoo-plankton, eggs and larvae of fishes.81

For what stated above, it is reasonable to look only for a statistical agreement, in the82

most favorable cases, between simulation and observation, since the evolution of a single83

trajectory strongly depends, in general, on the initial conditions and on the details of84

the flow. This means that a good model should be able to simulate the time evolution85

of average quantities characterizing Lagrangian motion like, e.g., net displacement of a86

tracer concentration, growth of the variance of the particle distribution, etc. An inter-87

esting approach to this problem consists in the use of data assimilation techniques, in a88

Lagrangian context, which allow some significant improvement in the reconstruction of89

unobserved tracer trajectories, starting from the knowledge of a representative sample90

[Ozgomen et al., 2000; Piterbarg , 2001].91

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Information on ocean currents are usually provided by numerical model outputs, for92

which one must assume, of course, that the reconstructed velocity fields are not perfectly93

realistic, but contain errors. The validation of the model forecasts and the quantification94

of the errors associated to the forecast fields are challenging problems.95

In this paper, we consider the Mediterranean Forecast System, or MFS, as ocean model96

providing Eulerian velocity fields used to simulate numerical (surface) drifter trajectories97

to be compared with ocean drifter data available for the Mediterranean Sea; FSLE-I is98

measured for both model and experimental data, in order to quantitatively estimate the99

dispersion rate of Lagrangian trajectory pairs in each case, separately; FSLE-II between100

model and data is measured in order to identify the separation scale (if any) beyond101

which numerical and real drifters depart from each other with same dispersion rate as any102

other model trajectory pair, i.e. when trajectory evolution becomes weakly sensitive on103

the differences between model and real dynamics; a 2D kinematic Lagrangian model, or104

KLM, is considered as parameterization of the velocity field components poorly resolved105

by MFS because of finite space and time resolution; FSLE-I and FSLE-II are, again,106

measured for the coupled model, MFS+KLM, in order to check if, and how much, the107

contribution of the subgrid kinematic field helps to simulate more realistically the actual108

drifter dispersion.109

The FSLE-I analysis of Lagrangian relative dispersion has become by now a popular110

research tool after it has been introduced and adapted to various applications of geophys-111

ical interest, e.g. drifter dispersion in the Adriatic Sea [Lacorata et al., 2001], mixing rate112

maps and barriers to transport [Boffetta et al., 2001], sensitivity to spatial resolution of113

an ocean model for the Mediterranean Sea [Iudicone et al., 2002], turbulent dispersion in114

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the stratosphere from experimental data [Lacorata et al., 2004], large-eddy simulations of115

planetary boundary layer turbulence [Gioia et al., 2004; Lacorata et al., 2008], etc.116

Recently, this technique has been widely applied to model and drifter analysis of ocean117

systems. For example [Poje et al., 2010] studied the sensitivity of an ocean model to the118

FSLE metric; [Schroeder et al., 2012] analyzed model errors in the submesoscale regime119

for coastal problems; [Haza et al., 2010] used the FSLE technique to measure dispersion120

characteristics from HF coastal data; [Poje et al., 2014] used results from a large drifter121

experiment to quantify errors in a state-of-the-art prediction model; finally [Haza et al.,122

2012] use this metric to develop stochastic parameterizations for submesoscale regime123

relative dispersion.124

At this regard, we would like to stress that the FSLE-II additional analysis and the125

deterministic KLM calibrated to the actual mesoscale dispersion, presented in this work,126

constitute original contributions to the discussion.127

This paper is organized as follows: in section 2 the FSLE-based methodology is recalled;128

section 3 contains the description of drifter and model data; in section 4 the mesoscale129

Lagrangian kinematic model is introduced; the results we have obtained are reported in130

section 5 and discussed in section 6.131

2. Error growth rate: the FSLE

In general, it is assumed that the mean growth rate of the distance between two tra-132

jectories, in the phase space of a dynamical system, is a function of the separation size.133

This quantity can be measured by the Finite-Scale Lyapunov Exponent, or FSLE, i.e. a134

generalization of the maximum Lyapunov Exponent [Ott , 2002] to non infinitesimal errors135

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[Aurell et al., 1996, 1997]:136

λ(δ) ≡ log ρ⟨τ(δ)

⟩ (3)137

In (3), τ(δ) is defined as the perturbation amplification time, to grow from δ to ρδ,138

and ρ ∼ O(1) is the amplification factor. The average ⟨·⟩ is meant over an arbitrarily139

large number of realizations, sampling all possible initial conditions in the phase space.140

We will agree to call λ(δ), defined by (3), first-kind FSLE, or FSLE-I, if δ is the distance141

between two (homogeneous) trajectories in the phase space of one dynamical system, i.e.142

δ ≡∥∥∥x(1) − x(2)

∥∥∥ for system (1) or δ ≡∥∥∥y(1) − y(2)

∥∥∥ for system (2) ; second-kind FSLE,143

or FSLE-II, if δ is the distance between two (heterogeneous) trajectories evolving in two144

different dynamical systems, sharing the same phase space, i.e. δ ≡ ∥x− y∥.145

In case of Lagrangian trajectories, the FSLE-I measures the scale-dependent relative146

dispersion rate [Boffetta et al., 2000b; LaCasce, 2008]. The FSLE provides quantitative147

information about the dominant physical mechanisms acting at various scales of motion,148

e.g. chaos, turbulence, diffusion. To this regard, it is worth recalling here some useful149

scaling rules. In case λ(δ) ∼ δ−ν , depending on the exponent, it is customary to associate150

relative dispersion to a specific regime [Boffetta et al., 2000b]: exponential separation [Ott ,151

2002] for ν → 0, Richardson’s turbulent diffusion [Richardson, 1926] for ν = 2/3, Taylor’s152

standard diffusion [Taylor , 1921] for ν = 2, ballistic separation, or shear dispersion, for153

ν = 1. These scaling laws will be useful later to interpret the results of the FSLE analysis154

of both first and second kind. We remark also that, on the basis of dimensional arguments,155

it can be useful to define a finite-scale relative diffusivity as K(δ) ∼ δ2λ(δ), i.e. a quantity156

which measures the turbulent diffusion coefficient as function of separation between two157

tracer trajectories.158

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In the next section we will describe the data we have used for our FSLE analysis.159

3. Lagrangian data

3.1. In-situ drifter data

Mediterranean drifter data, analyzed in this work, are provided by MyOcean160

(http://www.myocean.eu, product: INSITU MED NRT OBSERVATIONS 013 035),161

where a more detailed description of the available data set can be found. A subset of162

713 drifters, deployed in the Mediterranean Sea during the period 1990-2012, see Fig.163

1, have been selected for the analysis on the basis of quality control flag, duration and164

regularity in time of the trajectories. We consider this sample as representative of the165

actual sea surface circulation, against which numerical simulations will be checked.166

3.2. The Ocean Model

Model Lagrangian trajectories are simulated by Mediterranean Forecasting System167

(MFS) reanalysis velocity fields, available as MyOcean (Ocean Monitoring and Forecast-168

ing) products. (MEDSEA REANALYSIS PHYS 006 004 myov04-med-ingv-cur-rean-dm)169

MFS is a hydrodynamic eddy-permitting model with a variational data assimilation170

scheme for temperature and salinity vertical profiles, and satellite Sea Level Anomaly171

along track data. The Mediterranean OGCM (Ocean General Circulation Model) code172

is NEMO-OPA (Nucleus for European Modelling of the Ocean-Ocean Parallelise) version173

3.2. NEMO has been implemented in the Mediterranean at 1/16◦ x 1/16◦ horizontal174

resolution and 72 unevenly spaced vertical levels [Oddo et al., 2009].175

The model is forced by momentum, water and heat fluxes interactively computed by176

bulk formulae using the 6 hours, 0.75◦ horizontal resolution ERAInterim reanalysis fields177

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from the European Centre for Medium-Range Weather Forecasts (ECMWF) and the178

model predicted surface temperatures (details of the air-sea physics are in Tonani et al.179

[2009]). The data assimilation system is the OCEANVAR scheme developed by Dobri-180

cic and Pinardi [2008]. The background error correlation matrix is estimated from the181

temporal variability of parameters in a historical model simulation. Background error182

correlation matrices vary seasonally and in thirteen regions of the Mediterranean Sea,183

which have different physical characteristics. The assimilated data include: sea level184

anomaly, sea surface temperature, in situ temperature profiles by VOS XBTs (Voluntary185

Observing Ship-eXpandable Bathythermograph), in situ temperature and salinity profiles186

by argo floats, and in situ temperature and salinity profiles from CTD (Conductivity-187

Temperature-Depth). Satellite OA-SST (Objective Analyses-Sea Surface Temperature)188

data [Buongiorno Nardelli et al., 2013] are used for the correction of surface heat fluxes189

with the relaxation constant of 60 W/m2K.190

MFS model re-analysis are daily fields covering the period 1987-2012. The re-191

analysis have been initialized with a gridded climatology for Temperature and Salin-192

ity computed from in-situ data sampled before 1987 (PRE-TREANSIENT climatol-193

ogy) from SeaDataNet FP6 project. The model has been initialized at the 1st Jan-194

uary 1985. The assimilation of the available satellite and in situ data is done since195

January 1st 1985 too. Two year of spin-up are considered, thus the available data196

starts in 1987. For more details on the procedures adopted by the data producers see197

http://catalogue.myocean.eu.org/static/resources/myocean198

/pum/MYO2-MED-PUM-006-004-V4.0.pdf199

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In this paper the MFS daily current fields covering the period 1990-2012 were used.200

The quality of the Med-reanalysis system has been assessed for the full time series in201

the framework of MyOcean. The model has been validated by computing the accuracy202

of SST, Temperature, Salinity and SLA forecasts against in situ and satellite data. The203

observations are the same used for the assimilation, defined semi-independent, but the204

statistics is computed before the assimilation correction is applied to the analysis [Tonani205

et al., 2009]. The results showed that the Mean SST RMS is 0.56 ◦C characterized by a high206

seasonality, SST RMS errors along the water column is on average 0.35 ◦C with surface207

RMS around 0.5 − 0.6 ◦C, RMS errors on the salinity field along the water column is on208

average 0.1 psu with maximum RMS at the surface of about 0.3 psu. The surface current209

velocity field has not been validated against in situ current observation. An indirect210

information on the quality of the surface velocity field has been made by comparing the211

model SLA with respect on satellite SLA. The results showed that he SLA model error212

at basin level or the order of 3.5 cm (see for the detail the MFS QUiD available from213

MyOcean)214

4. Subgrid parameterization: the kinematic Lagrangian model

There are various techniques available to describe subgrid motion in numerical models.215

In situations where the large-scale flow is much more energetic than the small-scale compo-216

nents, for example, an elegant approach based on perturbative results from multiple-scale217

expansions allows an accurate description of absolute dispersion observables [Mazzino,218

1997; Mazzino et al., 2005; Cencini et al., 2006]. In our case, we are mainly interested in219

modelling relative diffusion and we choose to adopt a different strategy.220

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Mesoscale turbulent pair dispersion is simulated by means of a kinematic Lagrangian221

model, or KLM, i.e. a deterministic velocity field, analytically defined in terms of spatial222

derivatives of a given stream function, which gives rise to chaotic Lagrangian trajectories223

[Lacorata et al., 2008; Palatella et al., 2014]. At this regard, we recall that the lack of mo-224

tion of the Eulerian structures of the kinematic velocity field, at the origin of the so-called225

“sweeping” problem [Thomson and Devenish, 2005], can be overtaken by computing the226

components of the kinematic velocity field in the reference frame of the mass center of227

a particle pair [Lacorata et al., 2008]. Since, in ultimate analysis, we are interested in228

modeling surface drifter motion, we will define the KLM as a 2D multi-scale lattice of229

(horizontal) convective cells. The resulting velocity field is a superposition of N inde-230

pendent spatial modes, incompressible, nonlinear and explicitly time dependent, so that231

Lagrangian trajectories generated by each of the spatial modes are generally chaotic on232

the corresponding scale of motion:233

uKLM(x, y, t) =Nm∑n=1

An sin[knx− knεn sin(ωnt)]×234

cos[kny − knεn sin(ωnt+ θn)]235

(4)236

vKLM(x, y, t) = −Nm∑n=1

An cos[knx− knεi sin(ωnt)]×237

sin[kny − knεn sin(ωnt+ θn)]238

Variables x and y are the spatial coordinates of a tracer particle; An and kn are velocity239

amplitude and wavenumber, respectively, of the spatial mode n; Nm is the number of240

modes; εn and ωn are amplitude and pulsation, respectively, of the n mode time oscilla-241

tions; θn are arbitrary phases. The velocity field defined in Eq. (4) is a generalization of242

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AB-type flows, well known in literature as simplified models of Rayleigh-Benard convec-243

tion [Solomon and Gollub, 1988; Crisanti et al., 1991]. Turbulent dispersion, e.g. particle244

pair separation growing in time according to the Richardson’s law [Richardson, 1926]245

in a 2D inverse cascade scenario [Kraichnan, 1967; Charney , 1971], can be simulated246

by assigning the Kolmogorov scaling [Frisch, 1995] to the velocity field amplitudes, i.e.247

A2n ∼ (ϵkn)

−2/3, where ϵ is the equivalent mean turbulent dissipation rate. The exact set248

up of the kinematic model will be described later.249

5. Results

FSLE-I for the drifters has been computed according to the following procedure: all250

available data are considered; a total number of ∼ 104 trajectory pairs are identified;251

a pair is analyzed in terms of FSLE only if, at a given time, the relative separation is252

less than a given threshold (5 km); possible time gaps in the trajectories are filled by253

polynomial interpolation; trajectories are discarded if there is a time gap wider than 3254

days in their record; a pair is analyzed as long as two trajectories are simultaneous. The255

mean lifetime of simultaneous drifter pairs is of order of two months. Eventually, in the256

model simulations, we will have to consider trajectory pairs having the same lifetime as257

the real drifter pairs. Results of the (first and second kind) error growth analysis are258

summarized in Fig. 2.259

FSLE-I for MFS simulations has been computed on 5 · 104 neutrally buoyant trajectory260

pairs, initially distributed on a 3 m-depth layer all over the domain, having initial sepa-261

ration ≃ 5 km, and covering the period from January to March 2009. The choice of the262

layer depth fulfills two requirements: to be sufficiently close to where real drifter motion263

occurs and, at the same time, significantly below the very first vertical level of the model264

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where wind forcing, and other boundary interactions, might alter the reconstruction of the265

currents. We have checked that the results are not significantly sensitive to the change of266

year for the model simulations. The plateau of the curve, i.e. the absence of a Richardson267

regime, indicates that, on scales below ∼ 50 km, MFS velocity field structures cannot268

adequately contribute to the relative dispersion process.269

This is in good agreement with the information derived from the Eulerian spectra cal-270

culated in [Skamarock , 2004]. This author shows that the dynamical structures are accu-271

rately defined on a scale at least ∼ 7 ·∆x, where in our case ∆x ≃ 6.5 km is the spatial272

resolution.273

In order to compensate this mesoscale “dynamical gap”, KLM is used as replacement274

of the velocity field components not well resolved by MFS. Taking advantage of the in-275

dications coming from the drifter analysis, the kinematic model has the following set up:276

ln = 2−1/2 ln−1, An = C0 (ϵln)1/3, εn = 10−1ln, ωn = 2πAn/ln, θn = π/4, where ln = 2π/kn,277

for n = 0, 1, ..., Nm, are the spatial wavelengths of the model; Nm = 6 is the number of278

modes; l6 = 10 km and l0 = 25/2l6 km are minimum and maximum scales, respectively,279

of the kinematic “inertial range”; ϵ = 10−9 m2 s−3 is the equivalent mean turbulent dis-280

sipation rate; C0 ∼ O(1) is a fine tuning parameter. This kind of parameterization of281

the unresolved, or poorly resolved, modes of the MFS velocity field allows to establish282

a Richardson scaling within the equivalent inertial range of the kinematic model, and,283

therefore, to reduce significantly the discrepancy with the experimental results.284

Seasonal variations of the dispersion properties displayed by the drifters (not shown)285

do not affect the shape of the scaling law but only change the value of the turbulent286

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dissipation rate ϵ by at most a 20% fluctuation with respect to the mean (positive in287

winter, negative in summer).288

From a practical point of view, let us examine what type of information the curves289

shown in Fig. 2 can provide. Let us assume a given tracer distribution to have an initial290

width of size δ0. Once the growth rate of the tracer variance is known (FSLE-I), one can291

estimate the mean time needed to increase the separation between two particles from δ0292

to δn = ϱnδ0 (with ϱ > 1) as follows. If the relative dispersion rate evolves according to a293

given scaling law, e.g. λ(δ) = c · δ−a ( with a, c > 0), then, recalling the FSLE definition294

(3), we can define a hierarchy of growth times:295

τ (n)(δ0) =n∑

m=1

τ(ϱm−1δ0) =log ϱ

c

n∑m=1

(ϱm−1δ0)a (5)296

such that τ (1)(δ0) ≡ τ(δ0) is simply the time needed to amplify δ0 by a factor ϱ, and τ (n)297

is the total time needed to amplify δ0 by a factor ϱn. Since the amplification time between298

two separation scales is proportional to the integral of the inverse FSLE, τ(δ) ∝ 1/λ(δ), it299

is evident in Fig. 2 what is the difference between a flat FSLE (MFS without KLM) and300

a Richardson-like scaling FSLE (drifters and MFS+KLM) in terms of dispersion times.301

For example, applying formula (5) to the results shown in Fig. 2, we can argue that, a302

pollutant concentration of initial size ∼ 10 km will take about 30 days to spread over a303

scale 8 times larger, according to MFS, and about 17 days, according to MFS+KLM and304

to drifter data. For scales larger than ≃ 80 km, the dispersion rate is the same for both305

cases.306

As far as second-kind FSLE analysis is concerned, the error growth between real data307

and numerical simulations has been measured according to the following strategy. We308

used MFS daily velocity fields available for the years from 2006 to 2009, and selected all309

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drifter data covering the same period. Every simulated trajectory has initial conditions310

lying on a real drifter track, so that δ = 0 at time t = 0 for every second-kind pair. In311

particular, a new numerical trajectory originates from a point located on a given drifter312

track each five day long time interval. The mean square separation between model and313

experimental trajectories is then measured, both as function of the scale, Fig. 2, and as314

function of time, Fig. 3. The error growth displays a ballistic-like regime for small errors,315

and a diffusive-like regime for large errors. This is true for both cases, MFS vs Drifters316

and MFS+KLM vs Drifters, using either the scale δ or the time t as independent variable.317

FSLE-II results are weakly sensitive to the small-scale details of the circulation model318

velocity field, i.e. the external kinematic contribution does not alter significantly the319

error growth rates measured between MFS and drifter trajectories. The existence of a320

threshold scale, on the other hand, beyond which FSLE-II and FSLE-I overlap, indicates321

that, as far as the relative dispersion process is concerned, the differences between the322

model velocity field and the actual sea surface circulation become negligible on sufficiently323

large separation scales. Even in this case, using Eq. (5), it is possible to estimate the324

time needed for this condition to establish.325

6. Discussion and conclusions

Reanalysis velocity fields, provided by the Mediterranean Forecasting System, have been326

used to test numerical simulations of Lagrangian transport and diffusion of passive tracers327

dispersed in the marine waters. A large amount of drifter data, recorded during various328

oceanographic campaigns between 1990 and 2012, publicly available as MyOcean (Ocean329

Monitoring and Forecasting) product, offers the opportunity to check Lagrangian simula-330

tions against experimental observations. To this regard, we have addressed the so-called331

D R A F T September 19, 2014, 5:28am D R A F T

LACORATA ET AL.: LAGRANGIAN PREDICTABILITY X - 17

first and second kind predictability problem, as far as the Lagrangian transport process is332

concerned, adopting a dynamical systems approach. The error growth on a trajectory, or,333

in more abstract terms, the distance between two states in the phase space of a dynamical334

system, is measured by means of the Finite-Scale Lyapunov Exponent technique, i.e. the335

optimal indicator of relative dispersion on finite scales. We refer to FSLE-I or FSLE-II336

depending if we are considering the displacement between homogeneous trajectories be-337

longing to the same dynamical system, or between heterogeneous trajectories belonging338

to different dynamical systems, respectively. A simplified but efficient parameterization339

of unresolved, or poorly resolved, velocity modes is obtained by means of a deterministic,340

multi-scale, nonlinear and time-dependent kinematic model which allows, at least from a341

statistical point of view, to fill the “dynamical gap” in the mesoscale range of the large342

scale circulation model. The interpretation of the results of our analysis, described in the343

previous section, can be summarized as follows.344

FSLE-I of the drifter data: a regime compatible with Richardson turbulent dispersion,345

λ(δ) ∼ ϵ1/3δ−2/3, where ϵ is the mean turbulent dissipation rate, occurs in the range346

[10-100] km, followed by a diffusive-like cut-off, λ(δ) ∼ δ−2, at larger scales; by fitting347

the Richardson scaling to the data, the order of magnitude of the turbulent dissipation348

rate results to be ϵ ∼ 10−9 m2 s−3, in agreement with other estimates measured in the349

ocean [LaCasce, 2008]; the spatial correlation length of the velocity field is of the order350

of the size of the most energetic coherent structures, ∼ O(102) km, which are ultimately351

responsible of the eddy-diffusion process on basin scale. We would like to stress that,352

in this particular context, investigating about the physical origin of the δ−2/3 scaling is353

D R A F T September 19, 2014, 5:28am D R A F T

X - 18 LACORATA ET AL.: LAGRANGIAN PREDICTABILITY

outside the scope of this work, so that the inverse cascade scenario is assumed only as a354

plausible explanation.355

FSLE-I of the MFS model without KLM: relative dispersion rates are nearly scale-356

independent, i.e. λ(δ) ≃ constant, and lower than the analogous experimental values,357

within a range [10-50] km; for larger separation scales, MFS and drifters have similar358

FSLE-I, compatibly with the statistical errors; this fact suggests that MFS underestimates359

the separation rate of the trajectory pairs, as long as the separation size is smaller than360

≃ 50 km, i.e. a scale of the same order as the Rossby radius in the ocean, except for a361

∼ O(1) factor, and about ten times larger than the spatial grid step; MFS, on the other362

hand, reproduces with good accuracy the same dispersion rates as the real drifters on363

separation scales ∼ 100 km, or larger, for which the finite resolution effects of the ocean364

circulation model tend to become negligible.365

FSLE-I of the coupled model MFS+KLM: the relative dispersion rates are, now, very366

close to the analogous quantities measured from the experimental data, even in the367

mesoscale range (see Fig. 2); the agreement is both qualitative and quantitative. This368

means that, from an applicative point of view, a numerical experiment of tracer disper-369

sion, in the sea surface layer, returns back growth rates of the particle pair separation, i.e.370

the size of the tracer distribution, very close to what is observed analyzing the real drifter371

pair evolution; from a theoretical point of view, on the other hand, the additional kine-372

matic field offers the possibility to restore a regime of turbulent pair dispersion, namely373

the Richardson law, i.e. the Lagrangian counterpart of a Kolmogorov energy spectrum374

possibly related, in principle, to a 2D inverse cascade scenario, otherwise missing in the375

mesoscale range of the MFS ocean model.376

D R A F T September 19, 2014, 5:28am D R A F T

LACORATA ET AL.: LAGRANGIAN PREDICTABILITY X - 19

FSLE-II: while FSLE-I measures the growth rate of the size (variance) of a tracer distri-377

bution, FSLE-II measures the mean displacement (error) between numerically simulated378

and actually observed drifter trajectories; at an early stage, the separation between real379

and model trajectories grows, on average, linearly in time, with a mean drift relative380

velocity of the same order as the unresolved (or poorly resolved) mesoscale velocity com-381

ponents; this is true in both cases we have considered, i.e. MFS with and without KLM;382

while indeed, from a statistical point of view, the kinematic model helps the general circu-383

lation model to recover the same behavior of the FSLE-I measured from the drifter data,384

from a single trajectory perspective, instead, the presence of the additional velocity field385

does not lead to any significant improvement; for sufficiently large errors, on the other386

hand, i.e. when spatial correlations are smoothed out, two heterogeneous trajectories tend387

to depart from each other according to a diffusive-like regime, just like any homogeneous388

trajectory pair (either real drifters or numerical tracers) would do; the point where FSLE-389

II and FSLE-I rejoin identifies a threshold scale (≃ 100 km) beyond which, substantially,390

the mass centers of real and virtual tracer distributions depart from each other with a391

growth rate not larger than the growth rate of the variance of the single concentrations.392

We would like to stress a conceptual, more than technical, aspect of this problem. The393

fact that FSLE-II is weakly sensitive to the presence of KLM in MFS suggests that, for394

example, even nesting a high-resolution model into the MFS does not necessarily guar-395

antee a significant improvement of the Lagrangian forecast skills, from a single trajectory396

perspective, but can only have effects from a statistical point of view. In other terms,397

in order to get a much slower growth rate of the error between model and data, one398

should implement both a correct initialization and a suitable high-frequency data assimi-399

D R A F T September 19, 2014, 5:28am D R A F T

X - 20 LACORATA ET AL.: LAGRANGIAN PREDICTABILITY

lation scheme for the small-scale dynamics, parallel to the refining of the model resolution.400

Without considering, in any case, that a too high resolution becomes hardly sustainable,401

from a computational point of view, if one wants to explore tracer motion over a very402

large domain in a reasonable time.403

In general, one of the main challenges of the modeling community is the validation of404

OGCM forecasts, especially for operational systems which deliver data available to a large405

variety of users and applications. This paper demonstrates, for the first time, how a FSLE406

I- and FSLE-II-based dynamical system approach can be a method to validate the error407

associated to the forecast of velocity fields through the analysis of Lagrangian trajectories.408

Since in situ velocity data are very rare or absent, this work suggests a methodology to409

quantify the predictability error associated to a forecasting model using Lagrangian in410

situ observations. We demonstrate that FSLE-I and II are a powerful tool to assess411

model predictability, to define the impact of the unresolved scale in the model forecasts412

and to quantify the error associated to the simulations. We also demonstrate that the413

addition of a kinematic Lagrangian model, calibrated against statistical characteristics414

of the Lagrangian field deduced by drifter observations, allow to recover, even though415

partially, the error due to the unresolved scales of the Eulerian model. We believe that416

this approach can be used in the future not only to qualify model predictability but,417

also, to quantify the impact of new model improvements on the forecasts, so that it418

could be adopted by the modeling community as a standard method to check the model419

predictability skills.420

Acknowledgments.421

D R A F T September 19, 2014, 5:28am D R A F T

LACORATA ET AL.: LAGRANGIAN PREDICTABILITY X - 21

This work has been financially supported by SSD PESCA and RITMARE Research422

Projects (MIUR - Italian Research Ministery). This study has been conducted using423

MyOcean Products.424

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Figure 1. Sample plot of the 713 drifter trajectories selected for the data analysis. Color

palette refers to the drifter lifetime in days; spatial coordinates are in longitude-latitude degrees.

Drifter data have been recorded during the period 1990-2012.

Figure 2. FSLE-I relative to: (⃝) drifter data; (2) MFS data only; (△) MFS+KLM data.

FSLE-II relative to: (▽) MFS vs drifters; (3) MFS+KLM vs drifters. For all curves, the

amplification factor is ϱ =√2. Reference scaling laws are: δ−2/3, Richardson turbulent diffusion,

and δ−1, ballistic or shear dispersion. Where not plotted, the statistical error bars are of the

same size as the symbols, or less.

Figure 3. Root mean square error, model vs data, as a function of time relative to: (⃝) MFS

vs drifters; (△) MFS+KLM vs drifters. Ballistic-like (∼ t) and diffusive-like (∼ t1/2) regimes are

approached in the short time and long time limit, respectively. In the inset a detail of the early

regime of the error growth is shown.

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10-1

100

101

102

FS

LE

λ

(δ)

(1/d

ay)

separation scale δ (km)

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DriftersMFS

MFS+KLMMFS-Drifters

MFS+KLM Drifters

100

101

102

10-1

100

101

102

err

or

∆(t

) (

km

)

time t (day)

t

t1/2

MFS - DriftersMFS+KLM Drifters

0

100

200

0 10 20 30

∆(t

) (

km

)

time t (day)