The effect of vertical shear on the horizontal dispersion properties
of passive tracer particles on the continental shelf of the South
Mediterranean is investigated by means of observation and model data.
In situ current measurements reveal that vertical gradients of
horizontal velocities in the upper mixing layer decorrelate quite fast
(

The role of small-scale motion in geophysical flows is
receiving renewed attention

Three-dimensional turbulence is thought to mostly
have an homogenizing effect, smearing sharp gradients and promoting
super-diffusive separation in time of initially close
trajectories. The relative eddy diffusivity is expected to grow as the

While the mathematical formulation of the
problem of turbulent dispersion can be considered established

From float trajectories analysis, Ollitrault
and collaborators (

More recently, Poje et al. (

When dealing with
ocean diffusion, there is a huge experimental gap between
buoyant/surface/two-dimensional processes and three-dimensional ones,
the former being much more observed than the latter. Lagrangian
diffusion due to horizontal velocity variations across the
three-dimensional structure of the mixing layer is clearly crucial to
the transport and fate of sediments, biological material such as
chlorophyll, and contaminants suspended in the ocean

To simplify the problem, one might be tempted to use depth-averaged currents for predicting horizontal dispersion, so neglecting vertical shear effects. As is discussed in what follows, this approach can be misleading and can have some important practical drawbacks when estimating the dispersion of 3-D tracers.

The effect of vertical shear on the horizontal
dispersion was first experimentally investigated by

From the
numerical modelling point of view, being able to simulate Lagrangian
dispersion in the ocean has great relevance, but it is a delicate task
because of the finite resolution of the circulation models, and more
fundamentally because of the nonlinear character of the
dynamics. Indeed, when dealing with basin-scale models, not only the
mixing layer dynamics is often missing, but also the velocity field
features from sub- to mesoscales are poorly resolved both temporally
and spatially. In this regard, various techniques

In this
paper, we focus on the role of vertical shear as an important mechanism
promoting the horizontal diffusion in the upper ocean. By vertical
shear, we mean the vertical gradients of the horizontal
velocities. The approach here considered consists in combining
observation and model data to assess the effect of vertical shear for
the tracer horizontal relative dispersion. Observation data come from
Acoustic Doppler Current Profilers (ADCPs) deployed in a narrow region
of the South Mediterranean. Numerical data come from the Mediterranean
Forecasting System (MFS) model, and are supplemented with the use of
deterministic kinematic models

(Colour online.) Three instances of the observed instantaneous
current profiles of the horizontal velocities, from the ADCP located
at 31.91

The Kinematic Lagrangian Model (KLM) here adopted can be 2-D, to better account for the horizontal dispersion due to mescoscale eddies, or 3-D, to simulate vertical turbulent-like motions in the ocean mixing layer. Both dynamics are often underestimated in GCMs. Although our primary interest is in the former situation, we will discuss both.

The paper is organized as follows. In
Sect.

We analyse the profiles of the horizontal velocities recorded with two
ADCPs working at

In situ measurements are compared to current data, at the same
locations and for the same period, extracted from MFS

Our
primary interest being the vertical shear, we adopted the following
procedure in the statistical analysis of ADCP current profiles:

we remove the mean velocity component from the
current measurements at different levels, obtaining

for each

the velocity gradient residual times series,

Log-lin plot of the velocity gradient autocorrelation
functions versus the time lag. All data refer to the

In Fig.

For MFS curves, the situation is rather different:
in one case, the curve never really attains zero; in the other case,
it does on a time lag

Beside the characteristic timescales, it is
useful to quantify the amplitude of velocity gradient
fluctuations. Figure

Lin-log plot of PDFs of the vertical shear component

However, if we compare daily averaged ADCP with MFS data, the cores of the unitary variance PDFs are very similar (not shown): this implies that at least for mean fluctuations, experimental ADCP and numerical MFS data account for dynamical behaviours having the same mean amplitudes.

We start by considering a neutrally buoyant tracer particle whose
position is given by the three-dimensional vector

When considering Lagrangian dispersion, the
problem is easily reformulated in terms of the time evolution of the
pair separation vector

The situation is different when e.g. particles have the chance to
experience for some time a mean vertical shear. If this is the case,
with

We discuss different sets of numerical simulations based on the velocity configurations of the MFS model, also supplemented by the use of the kinematic model to describe poorly resolved motions. Kinematic models can be adapted to the different dispersion regimes, namely exponential separation, turbulent dispersion, and standard diffusion. Their implementation hence depends on the specific dynamics and specific range of scales that one wants to describe. Here, we compute transport properties by introducing statistical Lagrangian motions for the mixing layer motions (3-D KLM), and separately for the poorly mesoscale motions (2-D KLM). By doing so, we demonstrate that (i) small-scale motions (due to the 3-D KLM) enabling tracer pairs to explore the whole mixing layer do not modify the MFS horizontal dispersion properties, due to the anomalous persistence of vertical gradients of the horizontal velocities in the MFS model, which overrides the small-scale fluctuations; (ii) differently, the horizontal relative separation resulting from the introduction of the 2-D KLM is fast enough to become dominant with respect to the anomalous shear effect produced by the MFS solution.

An illustration of the effect of vertical shear on the mean horizontal dispersion of two particles, P1 and P2, initially separated along the vertical direction.

Lagrangian numerical simulations are performed using as large-scale
velocities the zonal

In the following, we first describe the 3-D KLM which is
meant to account for the transport and mixing in the upper layer of
the ocean; then we introduce the 2-D KLM accounting for the
poor-resolution of mesoscale horizontal motions. More details on the
KLM definition and implementation can be found in

In compact form, the three-dimensional velocity field of the
KLM,

The explicit form of the velocity components of the 3-D model
then results as follows:

In order to simulate the
mixing-layer dynamical effect of a multiscale velocity field with a
turbulent-like behaviour, as is customary we superimpose

The
KLM for the unresolved mesoscale motions is
built up, in analogy with the 3-D model, in terms of an ensemble of
horizontal cells forming a 2-D regular lattice

The
explicit form of the 2-D sub-grid velocity is

We performed three series of numerical simulations releasing

(Colour online.) Log-log plot of the finite-scale Lyapunov
exponent

An elastic collision takes place when particles meet the domain
boundaries. Within each pair, particles start at the same latitude and
longitude position, but they are vertically separated: one particle
starts at

The three series of simulations are
characterized as follows:

Series I: the KLM velocity is absent and particles keep their initial depth unchanged throughout the entire simulation. This is quite far from realistic conditions; however, this numerical experiment is useful to quantify the effect of the vertical shear solely due to the mesoscale MFS model dynamics.

Series II: the 3-D KLM term is switched on, with the parameters
shown in Eq. (

Series III: this differs from the Series II due to the fact that
in addition to the 3-D KLM, the 2-D KLM model is also implemented:

The most natural way to quantify Lagrangian dispersion statistics is
in terms of the moments

The measure of FSLE consists of fixing a set of
threshold scales,

Here, since we want to compare how the horizontal diffusion is influenced by the different flow realizations, in the FSLE we consider horizontal separations only.

In Fig.

Note that the scale at which MFS surface tracers deviate from drifters is larger that the model resolution: this suggests that scale resolution is crucial for Lagrangian statistics.

How does vertical shear affect these results? Can the vertical shear substantially modify horizontal dispersion? We address these questions using numerical data from Series I, II, and III.

In Series I, the effect of the vertical shear onto the horizontal dispersion comes from the MFS model only. The associated FSLE curves clearly indicate that vertical shear is able to promote horizontal dispersion. Neutrally buoyant tracers moving at different depths experience velocity differences: as a result they start to separate already at very small scales.

In Series II, the 3-D KLM terms are switched on, and particles vertically explore the whole mixing layer. The obtained FSLE curve is very similar to that of Series I, and in particular it turns out to be slightly below the latter. This finding is somehow surprising since, thanks to the introduction of small-scale turbulent-like motions, tracer pairs can explore the whole mixing layer. However, the fluctuations of the 3-D KLM do not substantially modify the horizontal pair dispersion, and actually they make it slightly slower in the present case. This suggests that the dominant effect is the one associated with the MFS vertical shear.

In Series III, both the 3-D and the 2-D KLM are switched on. The resulting FSLE is larger than that of Series II at any scale. This means that the most important dynamical correction to the MFS model is that associated with the 2-D KLM. Indeed, the dispersion effect induced by the mesoscale eddies inserted in the 2-D KLM overrides any other horizontal dispersion effects, including the one associated with the anomalous persistence of vertical gradients of the horizontal velocities in the MFS model.

We can summarize the results of the numerical simulations as follows. By comparing the horizontal dispersion of the bare MFS model with Mediterranean surface data, one sees that actual drifter pair dispersion follows a turbulent-like behaviour, whereas modelled surface trajectories separate more slowly and at a nearly constant rate. A way to solve this mismatch is to add a 3-D kinematic model, enabling vertical shear mixing and promoting surface horizontal dispersion also. However, the adoption of the 3-D KLM only does not seem an appropriate choice, since vertical gradients of the horizontal velocities have an anomalous temporal persistence, resulting in a spurious shear dispersion. Indeed, such persistence does not seem to have a counterpart in observational data, and we interpret this as an artifact of the poor temporal resolution of MFS model.

On the other hand, adding a two -dimensional kinematic model, one finds that the anomalous shear dispersion effects become practically negligible, being hidden by the more energetic dispersion processes occurring at the mesoscale. Clearly, mesoscale eddies are not pure 2-D structures, but they have a certain vertical development in the mixing layer. This implies that mesoscale turbulent dispersion is not a property of the surface layer only, but belongs to a whole vertical range of ocean layers. By adding the 2-D KLM for mesoscale eddies, one realizes that the effect is to have an efficient dispersion that covers the one due to mean vertical shear.

Finally, it is worth recalling that, as
shown in

In this paper, we have discussed the effect of vertical shear onto the
horizontal pair dispersion of tracer particles in a Mediterranean Sea
model. Numerical simulations with the MFS model show that, differently
from drifters, pairs released at the same depth tend to exponentially
separate with a dispersion rate nearly constant over a wide range of
scales, up to the mesoscale. At larger scales (

Now, the question arises, whether the proper small-scale ocean model velocity field is able to simulate the horizontal dispersion of a tracer having a 3-D structure in the mixing layer and below. The solution to this problem is very difficult, mainly because experimental data of 3-D tracer dispersion are not easily available. Different modelling solutions can be adopted to account for different problems, depending on whether the mesoscale, submesoscale or small scale is the relevant range of scales in the dispersion problem, none of which is straightforward.

For the specific problem
of the effect of vertical shear, a different possibility, yet to be
explored, is to build up an ad hoc

We gratefully acknowledge useful discussions with G. Lacorata, who collaborated with us in the first part of the study. This work has been supported by SSD PESCA and RITMARE Research Projects (MIUR, the Italian Research Ministry). This study has been conducted using Marine Copernicus Products, that we acknowledge. The MFS current data are retrieved from MyOcean as MEDSEA_REANALYSIS_PHYS_006_004 myov04-med-ingv-cur-rean-dm. ADCP data were kindly delivered by the Italian Ente Nazionale Idrocarburi (ENI SpA). These data are collected as part of an industry-sponsored initiative and are currently proprietary. Technical support by Ing. F. Grasso at ISAC Lecce is kindly acknowledged. Edited by: L. Kantha