We characterize sea-ice drift by applying a Lagrangian diffusion analysis to buoy trajectories from the International Arctic Buoy Programme (IABP) dataset and from two different models: the standalone Lagrangian sea-ice model neXtSIM and the Eulerian coupled ice–ocean model used for the TOPAZ reanalysis. By applying the diffusion analysis to the IABP buoy trajectories over the period 1979–2011, we confirm that sea-ice diffusion follows two distinct regimes (ballistic and Brownian) and we provide accurate values for the diffusivity and integral timescale that could be used in Eulerian or Lagrangian passive tracers models to simulate the transport and diffusion of particles moving with the ice. We discuss how these values are linked to the evolution of the fluctuating displacements variance and how this information could be used to define the size of the search area around the position predicted by the mean drift. By comparing observed and simulated sea-ice trajectories for three consecutive winter seasons (2007–2011), we show how the characteristics of the simulated motion may differ from or agree well with observations. This comparison illustrates the usefulness of first applying a diffusion analysis to evaluate the output of modeling systems that include a sea-ice model before using these in, e.g., oil spill trajectory models or, more generally, to simulate the transport of passive tracers in sea ice.

Sea-ice motion can be viewed as a superposition of a mean circulation and
turbulent-like fluctuations

Single particle analysis (here referred to as diffusion analysis) is
particularly useful for characterizing long-term trajectories as it clearly
decomposes the motion into mean (predictable) and fluctuating (unpredictable)
parts

To simulate tracer transport, one can use continuous or discrete passive tracer models.
Continuous models are usually based on the following advection–diffusion equation:

These approaches have been widely used to study the spread of pollutant and
other tracers by eddy turbulence in the ocean and atmosphere

Another way to use passive tracer models is to replace the mean field

Before using one of these approaches one needs to answer a few questions: What are the right averaging scales to define the mean motion field? What are the statistical properties of the fluctuating part of the motion? Is there a transition between different regimes of diffusion? Are the mean and fluctuating parts of the motion correctly reproduced by the forcing field? When not, could this indicate that some processes are missing in the forcing velocity fields? If the fluctuating part is not well reproduced, could it be compensated by adding extra terms in the tracer equation? These important questions are not always answered before running tracer models forced by sea-ice velocity fields and this could strongly impact the validity of the studies based on such results.

This is of particular importance in the context of the increasing activities
in the Arctic seas (e.g., shipping, fishing, and oil and gas exploration and
exploitation), which enhance the risk of pollution in a region where
ecosystems are already under threat from the amplified effects of climate
change. The extreme conditions (e.g., presence of ice, extreme cold, high
winds and the polar night) and the long distance from well-equipped
facilities may hamper access to the polluted area for several months

In this paper, we demonstrate the utility of applying Lagrangian diffusion
analysis to sea-ice trajectories in the context of passive tracer modeling.
The analyses presented in this paper are restricted to winter conditions, as
this season has been identified as more critical for oil spill recovery
operations

In this section we present the theoretical framework of the diffusion
analysis by showing its application to the IABP dataset from 1979 to 2011 as
a reference. Results and theory are then compared and discussed in the
context of passive tracer modeling. This is the same analysis as that already
done by

Example of a 35-day long trajectory from a
buoy of the IABP dataset (thin red line) partitioned into a mean (thick black
line) and fluctuating (thin black line) parts using the method described in
Sect.

Buoy tracks from the IABP dataset for the winter periods 1979–2011 (left panel) and the corresponding number of buoys (middle panel) and records (right panel). The central Arctic domain is delineated by a blue line.

Figure

This includes all buoy data north of 70

The raw IABP buoy positions are sampled irregularly in time with a mean time
interval of 1 h, and with errors ranging from 100 to 300 m, depending on the
positioning system they used

We manually checked each individual buoy track from the IABP dataset to clean
them from unrealistic “jumps” or “spikes” in the trajectories and from
obvious errors of the dating system. The unrealistic “jumps” present in
buoy trajectories are due either to errors in the positioning system
installed in the buoy or to wrong recordings during the deployment or
recovery phase of the instruments. A polar stereographic projection is used
to change the IABP and virtual buoy positions from geographic to Cartesian

The IABP buoy trajectories are geometrically complex, with abrupt changes in
direction (Fig.

Therefore, it is helpful to decompose the total sea-ice motion into a mean
part, which should be considered homogeneous and stationary, and a fluctuating
part, which should contain the unpredictable, local or non-stationary motion.
This is done here by following the classical approach used to study
Lagrangian particle trajectories

From the list of positions

The mean velocity field

In this study we use constant weights, meaning that the mean velocity is
simply defined as an average of all the 12-hourly velocities available in the
dataset that are within a circle of diameter

To verify that the averaging scales

Ensemble average autocorrelation function of the
fluctuating velocities computed from the IABP dataset for winter seasons 1979
to 2011 with three different averaging scales. The first zero-crossing time

Once the motion is decomposed into mean and fluctuating parts, it is possible
to analyze the diffusion properties in the medium by following the theory
developed by

For very long time intervals

Ensemble mean of the variance of the
fluctuating displacements

These two regimes are clearly detected in Fig.

Following Lagrangian turbulent theory, diffusivity is defined as

This table shows the total number of floats (Nrf),
the calculated integral timescale (

In the Brownian regime (with Eq.

To define the optimal averaging scales, we perform the decomposition of the
sea-ice motion and the diffusion analysis for scales ranging from

With these optimal averaging scales, the computed fluctuating displacements
variance

We also checked the stationarity of the variance of the fluctuating
velocities by comparing

This table gives an estimate of the search radii
and areas corresponding to 1, 2 and 3 standard deviations, respectively, and
for time horizons ranging from 1 to 30 days. These numbers are averaged over
the whole domain and period analyzed in Sect.

These evaluation steps ensure that the values given for

The first outcome of the diffusion analysis is to provide a simple and rigorous method to separate the mean circulation from the fluctuating motion, which can then be analyzed separately.

A second outcome is to quantify the diffusion properties of sea ice that can
then be compared to the diffusion properties of passive tracers in the ocean.
We note that the integral timescale

A third outcome of this analysis is to give a way to estimate the evolution
of the fluctuating displacement

Time evolution of the norm of the fluctuating
displacement

For forecasts longer than a few days, typically only the mean ice drift can
be trusted. Then, the long-term average standard deviation provided here
could be used to define the size of the search area around the position
predicted by the mean drift. For example, the search area could be defined as
a circular region with a radius equal to 3 standard deviations of the
fluctuating displacement. The search radius would then be about 84 km after
5 days (corresponding to a surface area of 22 200 km

The diffusion analysis may also be used to predict long-term (typically seasonal) sea-ice trajectories based on continuous or discrete tracer models. The average mean velocity needed by the tracer model may be defined from observations taken over the last few months, whereas the term reflecting the effects of the unpredictable fluctuations may be defined by the values of diffusivity and integral timescale derived from the diffusion analysis.

Compared to the analysis of the same data performed by

There are, however, some limitations when using the mean motion and mean
diffusivity to force passive tracer models:

the averaging smooths out local mean circulation features such as coastal currents;

the method is not well suited for studying dispersion as it assumes no spatial correlation;

the diffusivity could differ spatially and be not well represented by the basin-wide mean value;

the diffusivity could be affected by the long-term trend in the
mean speed identified in

Model output or reanalyses are often used to directly force passive tracer
models

In this section we compare observed trajectories from the IABP dataset to trajectories of virtual buoys (here called “floats”) whose motion is forced by sea-ice fields coming from two different model setups. Due to limited available computational time, this analysis is restricted to three consecutive winters. The period 2007–2010 has been selected for its relatively good data coverage, with more than 40 IABP buoys recording their positions simultaneously every day.

The float simulations are initialized at the same time and position as the
IABP buoys (280 individual floats). The positions of each float are sampled
every 12 h at the same time as the IABP buoys and stop when the IABP buoy
track stops or when the float enters into an area of simulated open water
(sea-ice concentrations less than 15 %). By doing so, three comparable
datasets with the exact same number of positions are obtained: (i) the
observed sea-ice trajectories, already discussed in Sect.

TOPAZ is a coupled sea-ice–ocean data assimilation system

The sea-ice–ocean model of TOPAZ (hereafter called the TOPAZ model) is used
here in free-run mode (i.e., no data assimilation is applied) in the same
configuration as in

The TOPAZ model in free-run mode has been evaluated in

The float tracking with the TOPAZ model is performed offline by using the hourly mean sea-ice velocity fields simulated by the model. The float-tracking system moves the floats with a simple Eulerian method. The virtual floats move in the quasi-homogeneous TOPAZ Arctic grid in order to avoid singularity errors at and around the North Pole that would arise with a regular longitude / latitude grid. The sea-ice velocities given by the TOPAZ model are interpolated with a bilinear method to the position of the virtual Lagrangian floats every hour. We checked that, for the timescale and spatial resolution considered here, this tracking method gives similar results to computing the float position during runtime with the advantage of remaining computationally efficient.

neXtSIM is a fully Lagrangian thermodynamic–dynamic sea-ice model, using an
adaptive finite element mesh and a mechanical framework based on the
elasto-brittle rheology

IABP buoys tracks (left) and their corresponding virtual tracks simulated by TOPAZ (center) and neXtSIM (right) for the winters 2007/2008 (top), 2008/2009 (middle) and 2009/2010 (bottom).

Mean sea-ice velocity field computed from the
IABP buoys dataset (left), the corresponding float dataset generated with
TOPAZ (center) and neXtSIM (right) for the winters 2007/2008 (top),
2008/2009 (middle) and 2009/2010 (bottom). The mean velocity vectors are
computed with the averaging scales

Probability density function of the mean speed of the IABP buoys (left) and of the corresponding virtual floats in TOPAZ (middle) and neXtSIM (right) for the period 2007–2010. The Gaussian (dotted lines) and exponential (dashed lines) fits of the data are also indicated.

Probability density function of the fluctuating speed of the IABP buoys (left) and of the corresponding virtual floats in TOPAZ (middle) and neXtSIM (right) for the winter periods 2007–2010. The Gaussian (dotted lines) and exponential (dashed lines) fits of the data are also indicated.

The configuration used here is the same as the one presented and evaluated in

The neXtSIM model is able to simulate correctly the observed evolution of the
sea-ice volume, extent and area for the freezing season (from September to
May) but simulates a too-rapid melt from May onwards

For the winter season 2007–2008, the simulated drift fields have been
extensively evaluated in

The float tracking with neXtSIM is performed at runtime. The main reason for
doing this is that the Lagrangian advection used in the neXtSIM model offers
some additional challenges to a post-processing approach using Eulerian
fields due to the remeshing technique applied (see

Figure

The mean velocity fields for each winter season and for the three datasets
are shown in Fig.

The statistical distribution of the mean velocity gives valuable information
and can be used to evaluate the simulated mean drift more objectively.
Figure

When removing the mean part of the velocity field we are left with the
fluctuating velocity field

The fluctuating speeds of the IABP buoys clearly follow an exponential
distribution with a mean equal to 6.98 cm s

The mean value of the fluctuating speeds from the TOPAZ setup is too high by
about 30 % (8.97 cm s

Ensemble mean of the variance of the fluctuating
displacement

Diffusivity fields obtained from the analysis of the IABP buoys trajectories (left), TOPAZ float trajectories (middle) and neXtSIM float trajectories (right) for the winters 2007–2010. The diffusivity is averaged over boxes of 400 by 400 km.

Mean ice thickness in the central Arctic obtained
from ICESat satellite observations

Figure

Figure

The diffusivity field computed from the IABP buoys is not uniform and seems
to be related to the spatial distribution of sea-ice thickness shown in
Fig.

The goal of the present analysis is not to compare the model systems
themselves but to illustrate how the simulated motion fields differ from
observations and what would be the impact of using such model outputs to
force passive tracers models to study, for example, trajectories of pollutant
trajectories in sea ice. The differences between the simulated and observed
motion may be due to many factors, ranging from the internal characteristics
of the sea-ice models (their rheology, drag parameterization, etc.) to
external causes, such as the initial conditions, atmospheric forcing and
impact of the ocean. To distinguish the effects of each factor would require
one
to run the same model with different initial conditions, forcings and set of
parameters or to run different models in the same configuration (initial
conditions, forcings, parameters, etc.). Other diagnostics than the
diffusion analysis would also be necessary. For example, the effect of the
rheology would be better analyzed by a dispersion analysis (double particle
diffusion) as in

The TOPAZ model reproduces the very basic characteristics of the Arctic
sea-ice mean circulation, with interannually varying Beaufort Gyre and
Transpolar Drift. However, the averaged mean drift is overestimated by about
48 %. This overestimation partly comes from missing the mean drift speed
smaller than 5 cm s

In the simulations analyzed here, TOPAZ misses both the low (larger than
10 cm s

The neXtSIM model also reproduces the interannually varying Beaufort Gyre and
Transpolar Drift. The statistical distribution of the mean circulation is
close to the one obtained from the observations for the range of 0 to
4 cm s

In the simulations analyzed here, neXtSIM represents well the statistical
distribution of the fluctuating speed until 30 cm s

The two model setups used here have a common deficiency at representing the
fluctuating speed higher than 30 cm s

It is common practice to add an extra diffusive term to the tracer evolution equation, as discussed earlier. In the case of the TOPAZ setup, adding such an extra term to the tracer evolution equation would not help, as the model already overestimates the fluctuation both in the ballistic and Brownian regimes. Adding such a term when using the neXtSIM setup presented here may improve the evolution of the fluctuating displacement in the ballistic regime. Adding a random term would increase the fluctuating velocity variance but decrease the integral timescale. This may allow us to maintain the good performance in reproducing the long-term displacements and diffusivity fields, without impacting the long-term mean drift.

In the first part of the paper (Sect.

This information can be used in the context of pollutant tracking to
evaluate the proper size for the search area around the long-term trajectory
predicted by the mean drift. If one defines the search area as a circular
region with a radius equal to 3 standard deviations of the fluctuating
displacement, which would include the polluted area with 99.6 % confidence,
we find that on average the search radius should be about 84 km after 5 days
(corresponding to a surface area of about 22 000 km

The estimates of the mean drift field, diffusivity and integral timescale computed here could also be used within a passive tracer model (either with an advection–diffusion equation or a Lagrangian stochastic approach) to estimate the probability for a particle to be in a given position after a given time. The limitations of that approach would be the excessive smoothing of local mean circulation patterns (e.g., coastal currents), the inability to represent dispersion and the potential misrepresentation of the spatial and temporal distribution of the diffusivity values.

In the second part of the paper (Sect.

The mean velocities in the simulations using TOPAZ are on average 50 % too high and generally miss the very low mean drift located near the Canadian Arctic Archipelago. The long-term displacement variance and absolute diffusivity are also overestimated by about 50 %. Using the output of this TOPAZ setup for tracer studies would produce too long trajectories and too large displacement variance, potentially affecting the conclusions of such studies.

The mean velocities in the simulations using neXtSIM are on average 15 % too low; they reproduce well the very low mean drift located near the Canadian Arctic Archipelago but miss the highest values of the mean motion. The long-term displacement variance and absolute diffusivity fit well the observations. Tracer studies based on such results could be trusted except for the ballistic regime (first few days), where the simulated displacement variance is too weak.

Using the outputs of the simulations made with neXtSIM would give better sea-ice trajectories than using the outputs of the TOPAZ simulations analyzed
here. However, whether this difference mainly originates from (a) the different
initial conditions, forcing and ocean or (b) the sea-ice model itself
(different rheologies and thermodynamics) cannot be clearly answered. As a
follow up of this study, it would be interesting to investigate the causes of
the missing high values of fluctuating velocities and the overestimation of
the integral timescale, first by examining the impact of the atmospheric
forcing resolution and second by checking how the inertial/tidal oscillations
are represented by the two modeling platforms used here. To better asses the
quality of the simulated sea-ice dynamics, it would be interesting to also
perform a dispersion analysis as in

We would like to acknowledge L. Bertino, J. Xie and P. Griewank for interesting discussions and their contribution to the development of TOPAZ and neXtSIM. We also thank T. Williams for his help improving the manuscript. This work was supported by funding from the Oil and Gas Producers (OGP), TOTAL E &P and the Research Council of Norway via the SIMech project (no. 231179/F20, 2014–2016).Edited by: D. Notz Reviewed by: T. Martin and two anonymous referees