Introduction
Large changes in the Arctic sea ice have been observed in recent decades in
terms of the ice thickness, extent and drift e.g.. These changes, and their underlying
driving mechanisms, still need to be fully understood in spite of their being
fundamental for building confidence in the forecasting capabilities of
current prediction systems. The need for a reliable sea ice prediction
platform is particularly important in the modern context of growing economic
opportunities with high societal and environmental impacts. For instance, the
dramatic decline of sea ice cover in the Arctic is opening new shipping
routes, fishing grounds and tourist destinations as well as access to a
significant portion of the remaining hydrocarbon resources. Associated with
this increasing activity are important risks for pollution of the Arctic
environment and to human lives. High-quality predictions of ocean and
sea ice in the polar regions are therefore needed in order to measure the
risks, to plan future activities and to assist operations in real time.
Current short-term (i.e. within 10 days) sea ice forecasting systems
integrate either a stand-alone sea ice model (RIOPS,
), a coupled ice–ocean model (e.g. ACNFS, , TOPAZ, or
GIOPS; ) or more seldom a coupled
atmosphere–ice–ocean model (GloSea5, ). Seasonal
to decadal climate forecasts are more common and include sea ice as part of
the Earth system models (see e.g. ). The sea
ice models used in these systems are usually derived from the work of
, and they treat the sea ice as a continuous medium with a
viscous–plastic rheology . In spite of this
development, simple free-drift ice (i.e. in the absence of friction and
internal forces) forecasts have remained in use by environment agencies
. The forecast skill of these systems based
on a free-drift ice has been evaluated in deterministic mode, when a single
“best” forecast is provided: despite the lack of realism in the free-drift
assumption, the forecast skill of such systems is seen as difficult to beat
.
Probabilistic forecasts, widely used in weather forecasting
, are still in their infancy in sea
ice forecasting. Probabilistic predictions rely on an ensemble of model
simulations (i.e. a Monte Carlo simulation) used to describe the forecast
uncertainty stemming from errors in the model parameters, initial and
boundary conditions, and any external forcing. The resulting cloud
of model outputs is used to retrieve statistical information, such as the
ensemble mean and its spread (i.e. the standard deviation), which are thus
used in place of the deterministic forecast and to estimate the associated
uncertainty, respectively. The multiple simultaneous sources of errors usually make
the forecast accuracy of the ensemble mean exceed that of the single
deterministic prediction , although often the spread
underestimates the actual forecast error when the sources of error are not
all adequately accounted for . Monte Carlo techniques are
already common practice in different areas
e.g.
and a common tool for sensitivity analysis.
This study concerns the probabilistic forecast capability of the sea ice
model neXtSIM . The work is carried out by
performing a Monte Carlo sensitivity analysis of the model with respect to
uncertainties in the surface wind velocity. The first goal is to highlight
the role of the ice rheology in the ice drift: how do the ensemble mean drift
and its standard deviation respond to uncertainties in the wind forcing? To
answer this question, we compare the ice drift obtained from neXtSIM
to one obtained from a free-drift model. In the second part, we
study the skill of the probabilistic forecast using Lagrangian trajectories
departing from independent virtual drifting buoys and compare them with real
observations without aiming to make it a key objective. We use the conceptual
framework of search and rescue operations where a probabilistic forecast is
commonly used to draw the search area of the ocean where drifting objects are
likely to be found . Contrary to
these studies, the present simulations are in the context of a
“hindcast–forecast”, using reanalysed atmospheric forcing fields but
assuming that they are affected by errors with the statistical properties
that could be expected from a numerical weather forecast in the Arctic. For
simplicity, we will use the word “forecast” instead of
“hindcast–forecast” throughout the paper.
Our main research tool and object of study is the sea ice model
neXtSIM. The model neXtSIM is based on a Lagrangian
numerical scheme and on a continuous approach using a newly developed
elasto-brittle (EB) ice rheology. This mechanical framework is inspired by the
scaling properties of sea ice dynamics revealed by multi-scale statistical
analyses of observed sea ice drift and deformation
() as well as by the in situ
measurements of sea ice internal stresses showing that sea ice deformation is
accommodated by Coulombic faulting (). For
40 years, a large variety of sea ice models have been developed. Some, like
neXtSIM, treat the sea ice as a continuous medium, but with
different rheologies (e.g. and modelled sea ice as
an elasto-plastic material; as an elasto-visco-plastic
material; and as an Maxwell elasto-brittle material),
and are suitable for high ice concentration (> 80 %), while others that
treat the ice as a discrete medium are more suitable for low ice concentration
(< 80 %) such as within the marginal ice zone.
We concentrate here on the impact of the error from the wind field alone. The
reasons are twofold. First, the wind is the most influential external force
affecting sea ice motion. About 70 % of the variance of the sea ice motion
in the central Arctic can be explained by the geostrophic winds
. However, the sea ice response to winds strongly
depends on its degree of damage; sea ice responds in a linear way only when
it is fragmented into small floes. Indeed, in this case, the internal forces
are negligible and the inertial term is linearly related to the air and water
drags, whereas this behaviour drastically changes when considering a large,
continuous and undamaged solid plate. The second reason is that surface
wind velocity fields provided by atmospheric reanalyses contain large
uncertainties in the Arctic due to the limited number of observations.
Previous sensitivity analyses of the neXtSIM model have been
performed with respect to initial conditions and to some key sea ice
mechanical parameters (see Sect. 4 in ). These analyses
consisted in running the model with different values of the input sources.
This allowed the authors to explore and quantify the sensitivity of the ice
velocity with respect to the ratio between water and air drag coefficients
and of the ice deformation with respect to the compactness parameter value
(see Eq. ), the sea ice cohesion value (see Eq. 10 in
), the initial concentration field or the initial
thickness field. Although these analyses did not use the latest model
developments of neXtSIM (in particular they included neither the
thermodynamics nor the re-meshing process), the impact of the mechanical
parameters on the ice deformation can still be considered as valid.
Systematic errors in the mean sea ice drift are evaluated by averaging
modelled and observed drift from the OSI-SAF dataset over
the period between 1 January 2008 and 30 April 2008 and over boxes of
100 × 100 km2 covering the whole Arctic (see
Fig. ). The largest differences between the observed
and simulated mean ice drift are located in the Beaufort, Chukchi, Kara and
Barents seas and Fram Strait and in some areas of the East Siberian Sea. In
the rest of the domain the error on the mean winter drift is only less than
3 km day-1, consistently with .
Systematic errors in the neXtSIM ice velocities compared to
observations from the OSI-SAF dataset. Simulated and observed ice drift is
averaged over the period from 1 January 2008 to 30 April 2008. The cells with
less than 28 observations over the winter are masked. The colour scale
represents the velocity in km day-1.
This paper is organised as follows: Sect. gives a general
presentation of the sea ice model neXtSIM with the main equations
describing the sea ice dynamical behaviour; Sect.
presents the details of the sensitivity analysis based on a Monte Carlo
sampling, including the description of the quantities of interest, the
construction of the wind perturbations, and the general experimental setup.
In the same Sect. , we also define the free-drift
model that will be used for comparison and benchmark against neXtSIM.
Section discusses the results for the ensemble mean, spread and
the evaluation of the forecast skills comparing neXtSIM to the free-drift
model. Final conclusions are drawn in Sect. .
General information on the model neXtSIM
In this section, we provide a general description of neXtSIM.
Deliberately, we choose to not go through all model equations here but
rather list those that are needed to get an overall understanding of how the
model works and that are relevant for the present study. For a more detailed
description of the model see and
.
neXtSIM is a continuous dynamic–thermodynamic sea ice model. It uses
a pure Lagrangian advection scheme, meaning that the nodes of the model mesh
are moving at each time step according to the simulated ice motion. The model
mesh is therefore changing over time, is not spatially homogeneous and
can locally become highly distorted, that is, when and where the ice motion
field is showing strong spatial gradients. In this case, a local and
conservative re-meshing procedure is applied in order to keep the numerical
integrity of the model and the spatial resolution of the grid approximatively
constant during the simulation. The equations are discretised on a triangular
mesh and solved using the classical finite element method, with scalar and
tensorial variables defined at the centre of the mesh elements and vectors
defined at the vertices. The model uses a mechanical framework that has
been developed recently () and is
based on the EB rheology. The brittle mechanical behaviour
of the sea ice is simulated by calculating the local level of damage in each
grid cell, a variable which is not considered in classical viscous–plastic
sea ice models typically used in the sea ice modelling community. Sea ice
thermodynamics, which is parametrised in neXtSIM as in the zero-layer
model of , controls the amount of ice formed or melted at
each time step. When a volume of new (and therefore undamaged) ice is formed
within a grid cell by thermodynamical freezing, the mechanical strength of
the total volume of ice covering that cell is partially restored, and the new
damage value is computed as a volume-weighted mean. Note, however, that the
damaging process is very fast (i.e. about few minutes) while the mechanical
healing process is occurring over much slower timescales of about several
weeks. The sea ice variables used in neXtSIM are the following: h
and hs are the effective sea ice and snow thickness,
respectively (ice and snow volumes per unit area); A is the sea ice
concentration (bounded to 1); d is the sea ice damage ranging from 0
(undamaged ice) to 1 (fully damaged); u is the horizontal sea ice
velocity vector; and σ is the ice internal stress tensor. The
model has two ice thickness categories: ice and open water.
The evolution
equations for h, hs and A (here denoted ϕ) have the
following generic form:
DϕDt=-ϕ∇⋅u+Sϕ,
where DϕDt is the material derivative of
ϕ, ∇⋅u the divergence of the horizontal velocity and
Sϕ a thermodynamical sink–source term. The evolution of sea ice
velocity comes from the following sea ice momentum equation, integrated over
the vertical,
mDuDt=∇⋅(σh)-∇P+τa+τw+τb-mfk×u-mg∇η,
where m is the inertial mass, P is a pressure term,
τa is the surface wind (air) stress,
τw is the ocean (water) stress and
τb is the basal stress in case of grounded ice
parametrised as in . The last terms are the Coriolis
parameter, f, the upward pointing unit vector, k, the gravity
acceleration, g, and the ocean surface elevation, η. The
internal stress σ is computed as in and
. Its evolution equation can be written as
DσDt=ΔdDt∂C∂d:ϵ+C(A,d):ϵ˙,
where d is the damage and ϵ˙ is the deformation rate
tensor defined as ϵ˙=12∇u+(∇u)T. C can be written as
C=E(A,d)1-ν21ν0ν10001-ν2,
where ν is the Poisson ratio and E(A,d) is the effective elastic
stiffness of the ice, which depends on the ice concentration A and the
damage d according to
E=Ye-α(1-A)(1-d),
where Y is the sea ice elastic modulus (Young's modulus) and α is
the so-called compactness parameter.
The evolution equation for the damage is written as
DdDt=ΔdΔt+Sd,
where Δd is a damage source term calculated as in
(Eq. 8), and Sd is thermodynamical sink term
which depends on the volume of new and undamaged ice formed over one time
step as well as on time (See , Sect. 2.3, for more
details).
The air and oceanic drags, respectively τa and
τw in Eq. (), are written as a force per unit
area in the quadratic form using the associated turning angle
τa=ρaCaua-uRθaua-uτw=ρwCwuw-uRθwuw-u,
where ., Rθa,
Rθw, ua, uw,
ρa, ρw, Ca and Cw are,
respectively, the Euclidean norm in R2, the rotation matrix
through the angle θa and θw, the wind
velocity, the ocean current, the air density, the water density, the air drag
coefficient and the water drag coefficient. The values of the model
parameters that are used for the simulations presented in this paper are
listed in Table .
Parameters used in the model with their values for the simulations
performed for this study.
Symbol
Meaning
Value
Unit
ρa
air density
1.3
kgm-3
ca
air drag coefficient
5.1×10-3
–
ca
air drag coefficient (for FD)
3.2×10-3
–
θa
air turning angle
0
∘
ρw
water density
1025
kgm-3
cw
water drag coefficient
5.5×10-3
–
θw
water turning angle
25
∘
ρi
ice density
917
kgm-3
ρs
snow density
330
kgm-3
ν
Poisson coefficient
0.3
–
μ
internal friction coefficient
0.7
–
Y
elastic modulus
9
GPa
Δx
mean resolution of the mesh
10
km
Δt
time step
200
s
Td
damage relaxation time
28
days
c
cohesion parameter
8
kPa
α
compactness parameter
-20
–
Sensitivity analysis
Methodology
In this study, we perform a sensitivity analysis using a statistical approach
based on Monte Carlo sampling of the model inputs. We focus on the response
of the model to the uncertainties in the wind velocity field. In particular,
we are looking at the response of sea ice drift to wind perturbations
representing these uncertainties. Our methodology is based on simulating
Lagrangian trajectories of virtual buoys using an ensemble run of the
neXtSIM model forced by slightly different (i.e. perturbed) wind
forcing (see Sect. for more details on the
generation of the perturbed winds).
The velocity of a given virtual buoy is calculated online, at each time
step, as a linear interpolation of the velocities simulated at the nodes of
the mesh element containing that buoy (see Lagrangian approach in
Sect. ). Each virtual buoy is associated with an initial
position x0∈D, with D being the initial domain, and a start
date t0∈Y, where Y is the time period of interest of this study (see
Sect. for more details). A buoy trajectory
is denoted g(x0,t0,t) with t∈[t0,T], where T
defines the duration of the individual simulations. For each initial position
x0 and start date t0, we simulate N trajectories
gii∈{1,…,N} from N model runs, each
one corresponding to a different realisation of the wind forcing. If a buoy
ends up in an ice-free element, it is then untracked further and its
trajectory discarded from the remaining analysis.
For each ensemble member (trajectory), we define the following Euclidean distances
∀i∈1,…,N,ri(t)=gi(x0,t0,t)-x0bi(t)=gi(x0,t0,t)-B(t),
where the quantity ri(t) is the distance of the member position at time
t, gi(x0,t0,t), from its departure origin,
x0=gi(t=t0). The second quantity, bi(t), represents the
distance between the member position at time t and the ensemble mean
position (i.e. the barycentre, B(t), of the ensemble), B(t)=1/N∑i=1Ngix0,t0,t, at the same time
t (see the top panel of Fig. ). We make use here of the
convention of using boldface for vectors and matrices and normal face for
scalar quantities; hereafter, we drop the explicit mention of the dependence
on x0 and t0, to simplify the notation.
Furthermore, we define a two-dimensional time-dependent orthonormal basis,
centred on B(t), whose axes are two perpendicular lines, one of which connects x0 to
B(t). The coordinates of
gi(t) on this basis are hereafter denoted as bi,∥(t)
and bi,⟂(t), as illustrated in the bottom panel of
Fig. ; they provide information on the spatial and
temporal evolution of the ensemble spread and shape and can also be used to
look at how the virtual buoy positions are distributed around the ensemble
mean over time.
From a 12-member ensemble of simulated trajectories of a virtual
buoy drifting during 10 days of which only two of them, denoted i and j,
are drawn, we represent the distances (a) r and b and the
coordinates (b) b∥ and b⟂ for the virtual buoy i
and j at time t.
With the individual ri and bi in hand, we compute basic, second-order
statistics. Let us consider their means, μr and μb,
μr(t)=1N∑i=1Nri(t),μb(t)=1N∑i=1Nbi(t),
and the standard deviations, σb∥ and σb⟂,
of the components b∥ and b⟂,
σb∥(t)=1N-1∑i=1Nbi,∥(t)2andσb⟂(t)=1N-1∑i=1Nbi,⟂(t)2,
as our main quantities of interest in the analysis that follows. We note that
the mean of bi,∥ and bi,⟂ is zero (being barycentric
coordinates) and do not appear in the calculation of standard deviations.
Throughout the rest of this paper, σb∥(t) and
σb⟂(t) are only used to compute the ratio
R(t)=σb∥(t)/σb⟂(t),
which provides a measure of the anisotropy of the ensemble spread of the
virtual buoys positions around the barycentre B of the ensemble.
It is finally worth observing that the two quantities, r and b, provide
complementary information: the former on the advective component of the
motion and the latter on its diffusive part. The ensemble mean distance
from the starting point, μr, is a statistical estimate of the distance
travelled by an ice parcel according to the ice advection properties of the
motion field, while μb is the (mean) spread relative to the
aforementioned distance and accounts for the diffusion properties of the
motion; see the top panel of Fig. .
Experimental setup
Our domain of study is the region covering the Arctic Ocean. While the coasts
are considered as closed boundaries, open boundaries are set at the Fram and
Bering straits (see Fig. ).
Maps showing the Arctic domain considered for this study. The red
lines are open boundaries, while the black coastlines are closed boundaries.
The starting points of the virtual trajectories simulated with the
neXtSIM and FD models are represented by the blue crosses.
The wind forcing is taken from the Arctic System Reanalysis (ASR)
. This reanalysis product provides wind speeds and
directions at 10 m, every 3 h, at a horizontal resolution of 30 km. No
turning angle has been applied (see Table ). For every
3-hourly wind field, we generate spatio-temporal correlated perturbations as
described in and then add them to the ASR wind field.
This procedure is identical to the one used to produce ensemble runs with the
coupled ocean–sea ice model TOPAZ and constitutes the
propagation step in the ensemble Kalman filter . The method
is designed such that the perturbed wind fields keep important
physical properties; that is, the wind perturbations are geostrophic
(gradients of random perturbations of the sea level pressure) and the wind
divergence is kept unchanged. They are built on random stationary Gaussian
fields, with a Gaussian spatial covariance function, dimensionalised by the
wind error variance and correlated in time. Time series of wind perturbations
are assumed to be red noise. For our study, we used a decorrelation
timescale of 2 days, a horizontal decorrelation length scale of 250 km and
wind speed variance of 1 m2 s-2. These values are
identical to those used in except for a reduced wind speed
variance to maintain a consistency with the ice rheology in neXtSIM.
Indeed, a larger variance leads to an excess of ice breaking up beyond the
physical behaviour expressed in neXtSIM.
Although the ensemble average of the perturbed u and v components of the
winds is equal by construction to the original winds provided by the ASR, the
wind speed is positively biased. The value of the air drag coefficient
(Ca in Eq. ) had previously been optimised in the
neXtSIM model when forced by the ASR following the method presented in
Sect. 3.2 and set to 7.6×10-3. We
applied the same method here to tune the value of Ca so that the
simulated ice drift compares best with the observed ice drift from the
OSI-SAF dataset . The optimisation is carried out at all
times between 1 January and 30 April 2008 but limited to the region where the
ice is in free drift (see Fig. ), i.e. where the
ensemble-averaged simulated ice velocity differs by less than 10 % from the
drift simulated by the free-drift (FD) model.
Figure shows the comparison, after optimisation of
the air drag coefficient, between the observed and simulated ice velocities.
As expected for a wind dataset positively biased in magnitude compared to the
original one, we found an optimised value for the drag coefficient
Ca=5.1×10-3, lower than the one used in
(7.6×10-3).
The ocean forcing comes from the TOPAZ4 reanalysis . TOPAZ4
is a coupled ocean–sea ice system combined with an ensemble Kalman filter
data assimilation scheme assimilating both ocean and sea ice observations. In
our simulations, we used the 30 m depth currents, to which we apply a
turning angle of 25∘, the surface temperature and salinity, and the
sea surface height, all provided as daily means with an average horizontal
resolution of 12.5 km, following .
Spatial distribution of the number of occurrences of free-drift
events between 1 January and 30 April 2008. The temporal sampling frequency
used is 1 day. These are the instances used for the optimisation of the air
drag coefficient.
Scatter plots for the two components, (a) x and (b) y, of the simulated (neXtSIM, x axis) and observed
drift (OSI-SAF dataset, y axis) after the air drag optimisation procedure.
The cumulative distribution of the ice velocity errors is shown in
panel (c).
Our analysis is based on two periods of the year 2008: from
1 January to 10 May and from 1 July to 20 September, representative of the
winter and summer conditions, respectively. We have intentionally studied them separately
because winter and summer are characterised by significantly different sea
ice mechanical regimes and therefore drift responses. During the winter, the
whole Arctic basin is covered by ice and its concentration is close or equal
to 100 %; that is, the internal stresses in the ice, and the corresponding
∇⋅(σh) term in Eq. (), become large and of
the same order of magnitude as the wind drag term. As a consequence, the ice
drift is (on average) much reduced. During the summer period, in contrast, the ice concentration is lower and the ice pack does not generally
reach the coasts, the ice internal stresses are much closer or equal to zero,
and the ice drift closer to a free-drift state (see text below). We
note, however, that the wind field perturbations are generated using the same
aforementioned procedure, for both the winter and the summer, and have thus
the same spatial and temporal properties.
We ran a total of 13 simulations in the winter and 8 in the summer during
successive, non-overlapping 10-day periods. Limiting the length of the
simulations to 10 days ensures that the sea ice state (thickness and
concentration) remains as realistic as possible in the free-drift simulation,
in which there are no physical limits to the amount of ridging and opening.
The starting positions are separated by 100 km and cover the domain
displayed in Fig. . All ensemble members start from
the same initial conditions extracted from a previous – deterministic –
neXtSIM simulation by run without
any perturbations of the winds. This concerns all sea ice variables: h,
hs, A, d, u and σ. We ran an ensemble of
12 members, each of them forced by the perturbed wind generated as explained
above. We performed (not shown) a convergence analysis of our results as a
function of the ensemble size from N=3 to N=20 and observed a
convergence from about N=10 with only minor changes for N≥12, and
we are thus confident that N=12 suffices to our purposes. From these ensemble
runs we simulated a total of over 96 000 (≃8000×12) virtual
buoy trajectories over the winter season and over 38 000
(≃3200×12) trajectories over the summer season. This dataset
was used to run the analyses described in Sect. and presented
at the 19th EGU General Assembly .
As already stated, we compared neXtSIM with the so-called
FD model, so that all
simulations that follow have been carried out for the two models.
neXtSIM (Eq. ( with all terms on its right-hand side
included) is our reference model. The FD model is equivalent to
neXtSIM except that it considers the following simplified version of
the momentum equation in which the terms related to the sea ice rheology, the
basal stress and the inertial term are neglected:
0=τa+τw-mfk×u-mg∇η.
In Eq. () the water and air drag forces, the Coriolis force and the
gravity force due to the ocean surface tilt are balancing each other. The FD
model is therefore analogous to the steady-state drift of an object at the
surface. We run the FD model with the same initial conditions as neXtSIM
except that d and σ are not used. The drag coefficient is also
optimised for the FD model at a value of 3.2×10-3, which is lower
than for neXtSIM, as expected. The optimisation method used for FD is the
same as for neXtSIM described above, except that the OSI-SAF drift vectors
are used everywhere.
Results
In this section, the notations <.>W and <.>S
correspond to winter and summer averages (i.e. over all the 13 and 8
simulation periods of 10 days), respectively. The notations <.>D
correspond to the spatial mean over the domain. When considering both
spatially and temporally averaged quantities, we use the notations
<.>W,D or <.>S,D.
Spatial patterns
Figures and show maps of mean drifting distance and
spread (see the definitions of μb and μr in
Sect. ) of the virtual buoys after t=10 days, averaged
over the 13 (winter) and 8 (summer) successive simulations. Similar results
are obtained for different time t∈[0,10] days (not shown). The pixels on
the maps correspond to boxes of 100×100 km2 centred on the
initial positions x0, where the virtual buoys have been deployed at
t0.
Mean over the winter period of μr(t) and μb(t) at t=10
days. The calculated values are represented by coloured squares centred on
the starting points x0 shown in Fig. .
Winter average of wind speed and ice thickness. Both maps are from
the neXtSIM simulations, but a similar thickness field and the exact same
wind speed field are obtained for the FD simulations.
Figures and are the counterparts of Figs.
and and show the average wind speed (left panel) and ice
thickness (right panel) for winter (Fig. ) and summer
(Fig. ). Note that both figures are relative to
neXtSIM, but the free-drift wind speed is identical (same
perturbations) and the ice thickness geographical pattern very similar; we
have thus not displayed them to avoid redundancy.
From Figs. and we see that neXtSIM gives a
smoother response to perturbed forcing than the FD model in terms of mean
advective drift μr and mean diffusive spread μb, in both winter and
summer. Indeed, we observe in neXtSIM a clear spatial coherency in
both the advection and diffusion of the ice buoys over the domain that is
almost absent in FD. We believe that this behaviour is related to the mean
ice thickness pattern and, to a lesser extent, to the mean wind speed pattern
(see Figs. and for winter and summer, respectively).
For neXtSIM, the smallest values for the mean of μr and μb
averaged over the winter time period are found in the area located north of
Greenland and the Canadian Archipelago, which is where the ice is the oldest,
thickest (>4 m) and mechanically the strongest and where the winds are on
average weaker as compared to the rest of the Arctic. However, in
the surrounding seas (i.e. Beaufort, Bering, Chukchi, Kara and Barents seas
from west to east), where the ice is thinner and the winds stronger, the
means
of μr and μb are larger. Note that in summer these correlations or
anticorrelations are even stronger, for example between the means of μb
and the ice thickness (see Figs. and ).
For FD, the mean values of μr are correlated to the wind speed in winter
and, to a lesser extent, in summer (left panels in Figs. and
). It is worth noting that, despite the presence of thick ice in
the north of the Canadian Archipelago and low winds, the ice is still
advected significantly, as opposed to what is obtained with neXtSIM.
Moreover, the spatial pattern of the mean of μb shows no spatial
coherence and resembles the random patterns from the wind perturbations. It
is clearly visible in summer, while in winter the sea ice thickness field
stays discernible. This may be due to the presence of the ice mass in
Eq. ().
Mean over the summer period of μr(t) and μb(t) at t=10
days. The calculated values are represented by coloured squares centred on
the summer starting points x0 (not shown).
Summer average of wind speed and ice thickness. Both maps are from
the neXtSIM simulations, but a similar thickness field and the exact same
wind speed field are obtained for the FD simulations.
In both winter and summer, the time-averaged response of μr and μb
to wind perturbations is overall lower in neXtSIM than in FD (except
in summer when μr is 7 % larger in neXtSIM). This can be
attributed to the ice rheology being turned on in neXtSIM, thus
acting as an additional filter on the momentum transferred from the wind to
the ice. In more detail, it is interesting to note that the magnitude of the
impact of the ice rheology is different depending on whether we consider the drift
distance by advection r or the spread distance by diffusion b and
consider the winter or the summer. On average over the winter,
μr(t)D and μb(t)D
are,
respectively, 21 and 52 % lower in neXtSIM than in FD at t=10
days, whereas over the summer μb(t)D is 21 %
lower. This large difference between the two distances, especially in winter,
is probably related to the high ice concentration making sea ice harder to
break up, and it keeps the members closer to each other. During summer, the ice
is generally much less packed and the physical and dynamical differences between
neXtSIM and FD have a lower impact. The 7 % larger values of
μr for neXtSIM are likely related to the optimisation of the
air drag coefficient that returned a larger coefficient for neXtSIM.
As expected, for neXtSIM we observe an increase of μr(t) of
about 51 % and μb(t) of about 69 % in summer compared to winter.
This behaviour differs drastically from the FD for which the values are
nearly the same for both periods, and it is presumably related to the
decrease in ice concentration due to the summer melting. The averaged sea ice
concentration over the whole domain in winter is about 0.99 while it drops
to 0.83 in the summer. In neXtSIM, this strongly influences the
mechanical behaviour of the sea ice since the effective elastic stiffness E
depends non-linearly on the ice concentration (see Eq. ). Assuming
no change in the average level of damage of the ice, a drop by 15 % of the
ice concentration between winter and summer implies a reduction of E by
96 %. This reduction of E leads in turn to a significant decrease of the
internal stresses within the ice, thus lowering the term ∇⋅(σh) in Eq. (), which makes the buoys' drift in
neXtSIM closer to the ones obtained with the FD model.
The absolute values of μr and μb obtained by our analysis reveal
that the advection part of the motion is in general larger than the diffusive
part, independently of the season under consideration. In FD the ratio
γ=μr(t)/μb(t) at t=10 days is about 4.5. In neXtSIM, however, the ice rheology acts to increase this ratio to 7. However,
this value presents a strong spatial variability depending on the local
thickness and wind speed. Where both are large, γ is large. For
example, such areas are observed in the Fram Strait in winter (γ>10)
and in the central Arctic in summer (γ>12). Where both ice thickness
and wind speed are small, γ is small. For example, this is the case
around the new Siberian islands in winter (γ<4) and close to the ice
pack edge in summer (γ<6).
Spatial and temporal properties of the ensemble spread
Figure shows the probability density function (PDF) of
b∥(t) and b⟂(t) at t=10 days for both
neXtSIM and FD and for winter and summer (see
Sect. ). The PDFs of b∥ and b⟂ for the
FD case are almost identical, and so we chose to only display one curve (black
dashed line). The first aspect to remark from Fig. is that
all distributions are unimodal and symmetric, suggesting that the 2-D shape
of the ensemble is symmetric around its barycentre B. However, we
notice that the ensemble is anisotropic in neXtSIM, i.e. the
distributions of b∥ and b⟂ differ substantially,
whereas it is close to isotropic in FD.
Probability density function (PDF) of b∥(t) (solid lines) and
b⟂(t) (dotted lines) at t=10 days for neXtSIM in the winter
(blue) and summer (red). The PDFs from FD are similar for summer and winter,
and for b∥(t) and b⟂(t), and are therefore shown as a
single black dashed line.
Figure shows the temporal evolution over 10 days of the
Arctic averaged ratio R (Eq. (), which defines the degree of
anisotropy of the ensemble spread (1: isotropic; >1: anisotropic). We
observe on the one hand that R is very close to 1 and relatively constant
over time in the FD model. On the other hand, it is systematically larger in
neXtSIM, especially in winter, and it also displays a certain
short-term variability. Here, again, we encounter the peculiar effect of the
neXtSIM mechanical response to the external forces, which is to
break up and deform along fractures that are dispersing the different members
of the ensemble along a preferential direction; such behaviour cannot be
reproduced by the FD. Note also that R is as large as 2 within the first
2 days for neXtSIM in the winter, and then it decreases
monotonically for t>2, still remaining very large (between 1.4 and 1.6
at t=10 days). This reveals that the ice will first tend to move compactly
along the initial fractures (identical for all members at t=0) away from
the origin, but it then starts to break and, after 2 days, the damage pattern
becomes significantly different within each member, leading to a more
isotropic ice dispersion away from the barycentre.
In Fig. , we show the maps of the R(t) values computed for each
ensemble of trajectories at t=10 days. These values are represented as
coloured squares centred on the starting point x0. We observe that
highest degree of ensemble anisotropy (R>1) is found north of Greenland and
Canadian Archipelago, where the ice is the thickest and the ice drift and
winds the lowest, in overall agreement with the interpretation of the
temporal evolution of R for neXtSIM in the winter, provided in
relation with Fig. . Globally, we observe a high (>1.5)
anisotropy close to the coasts that can be explained by the ice pressure
that counteracts sea ice motions towards the coasts (and the associated
dispersion as well). In summer, large stretches of the coasts are ice-free
and the increase of R is less visible. This is in contrast to the pattern
from FD. In absence of internal stresses, the pattern of the anisotropy
exhibits no spatial coherence and is similar in both winter and summer
periods. Furthermore, as already noticed from Fig. , the
values obtained for neXtSIM are systematically larger (by about
65 %) than for FD during the winter whereas only 8 % larger during the
summer. However, and remarkably, the values of R, and thus the anisotropy of
the ice drift, for neXtSIM exhibit marked spatial correlations that
are absent from FD.
Evolution of the spatial mean of R(t) from t=1 to t=10 days
for the winter (blue) and summer (red) periods for neXtSIM (solid
lines) and FD (dashed lines).
Means over the winter (top panels) and summer (bottom panels) of
R(t) at t=10 days. The values are represented by coloured squares centred
on the starting points x0. The spatio-temporal mean values (i.e.
domain averaged and seasonally averaged) are calculated for each model and
displayed above each panels for reference.
Spatial domain average of the variance of the distances bi as a
function of time from t=12 h to t=10 days, for the neXtSIM
(solid) and FD (dashed) models and for winter (blue) and summer (red). As the
results for winter and summer are identical in FD, only one curve is
plotted (black dashed line). The Brownian regime (slope =1) is reached by
neXtSIM during the winter, while in the other cases a
super-diffusive regime is obtained (slope =1.15).
Another important characterisation of the ensemble spread evolution can be
set by looking at the variance of the distance b between the virtual buoys
and the barycentre B over time. The goal is to identify the diffusion
characteristics of the ensemble, which can be interpreted in the framework of
the turbulent diffusion theory of . Similar Lagrangian
diffusion analysis has been applied to study the regimes of diffusion of
surface drifters in the ocean e.g. and, more
recently, of buoys fixed to the ice cover e.g.. In the analysis performed
here, the distance b to the barycentre of the ensemble corresponds to the
fluctuating part m′ of the motion m in the so-called
Taylor decomposition m=m‾+m′. Figure
shows the temporal evolution of the ensemble average of the distances bi
averaged over the Arctic domain D calculated form the buoy's tracks
simulated with neXtSIM and FD. We found that the ensemble spread
follows two distinct diffusion regimes, one for small time, t≪Γ, and
one for large time, t≫Γ, where Γ is the so-called
integral timescale . In neXtSIM, the
first regime we found for winter corresponds to the ballistic regime
where 〈bi2〉D∼t2, and the
second to the Brownian regime where
〈bi2〉D∼t. These results are in
agreement with the wintertime sea ice diffusion regimes revealed by applying
the Lagrangian diffusion analysis to the buoy trajectories dataset of the
International Arctic Buoy Programme in wintertime
and show that our experimental setup based on ensemble simulations forced by
perturbed winds does not alter the capability of the neXtSIM model
to reproduce these properties, as also shown recently in
for the same winter. However, we note that the
regime we obtain with neXtSIM for the summer 2008 is
super-diffusive, with 〈bi2〉D∼t1.15 for t≫Γ, and therefore in apparent contradiction with
, who found that sea ice follows a same Brownian regime in
both winter and summer when averaging over the period 1979–2007. We suggest
that this may actually be the fingerprint of a change in the dynamical
behaviour of sea ice in summer that occurred over the most recent years
(including 2008), in which the rheology plays a weaker role than it did
in the 1980s and 1990s. This is also supported by the results we obtain here
with the FD model that neglects the rheology and exhibits
super-diffusive regime for 2008, regardless of the season
considered.
Predictive skills of neXtSIM and of the FD models
We evaluate here how well the neXtSIM and FD models are able to
forecast real trajectories in hindcast mode. As a benchmark, we compare the
ensemble runs from both models to 604 (in winter) and 344 (in summer)
observed trajectories from the IABP dataset. The simulated trajectories of
both neXtSIM and FD are initiated on the same initial positions and
at the same time as the IABP buoys (displayed in
Fig. ); the positions of IABP buoys are known every
12 h. It is important to note that most of these buoys were deployed in
regions of thick and compact ice, the drift of which is largely influenced by the
sea ice rheology. Therefore, we expect the FD model to be less competitive
than if the comparison data had been uniformly distributed across the Arctic.
Maps showing the positions (blue
crosses;
603 during winter and 344 during summer) of the IABP buoy trajectory
dataset used in this study as starting point of the ensemble trajectory
simulations performed with the neXtSIM and FD models. The grey area
marks the presence of the sea ice during at least 10 consecutive days (the
length of the simulations) during the winter and summer periods.
As a metric for the models skill intercomparison, we use the
linear forecast error vector
e(t)=B(t)-O(t),
defined as the distance between the observed IABP buoy position,
O(t), and that of the ensemble mean, B(t) (see also Fig.
). The components of e(t) onto the orthonormal basis
centred on O (see Sect. , Figs.
and ) read e∥(t) and e⟂(t).
We complete this evaluation comparing results from both models with those
from a single deterministic forecast in order to verify the advantage of
probabilistic forecasts. In this case, we run neXtSIM with
parameters found in , except the air drag
coefficient has been re-tuned to Ca=6.5×10-3 and
unperturbed winds. For this new air drag coefficient, the same optimisation
process as the probabilistic case is used against the same observations.
Mean of the absolute forecast error e and vector components e∥ and e⟂ in the
directions along and across the mean trajectory, as a function of drift duration. neXtSIM is represented by solid lines, FD
is shown as dashed lines, and the deterministic runs with cross marks. Winter is in blue and summer in red. A positive e⟂ represents a drift to the left of the trajectory.
Figure shows the average norm of the forecast error,
e, and of its components, e∥(t) and
e⟂(t), as a function of time, for the experiments with
neXtSIM and FD and for both winter and summer. Results reveal that
the forecast error is smaller in neXtSIM than FD in both seasons. In
winter, the error of the FD model grows almost twice as fast as the error of
neXtSIM, up to 26 km at day 10 compared to about 15 km for
neXtSIM. As already deduced from the results in the previous
section, the mechanics underlying of the ice drift in neXtSIM and FD
are similar in the summer, and this is reflected by the two errors being much
closer to each other: the difference between the two increases slower,
reaching ≃3 km after 10 days (see the left panel in
Fig. ).
The centre panel in Fig. shows a positive bias of the
error in the along-drift component (e∥) for both models and both
periods except for neXtSIM in winter, which presents a negative
bias. The general positive biases betray a too-fast drift in the direction
along the ensemble mean drift compared to the observations. Nevertheless, the
bias for winter in neXtSIM is 2.5 times smaller than in the FD
model, whereas both models perform similarly in the summer.
Finally, the right panel in Fig. also reveals a bias of
the error in the direction across the ensemble mean drift, that is substantially
weaker than in the previous case. For FD, e⟂ still being negative for
both periods corresponds to a drift too far to the right compared to the
observations. This bias to the right could be further reduced by a separate
tuning of the turning angle θw for the FD and
neXtSIM models.
Overall, we conclude that the performances are significantly better for
neXtSIM in winter but similar in summer, and this would likely
remain so even after optimal tuning of the turning angle.
Comparing to a single deterministic neXtSIM forecast, we note that
the forecast error is close to the average of the probabilistic run but
larger in summer, reaching 34 km at 10 days. The main difference with the
probabilistic run is the poorer along-drift component e∥. Indeed,
the error is closer to zero in winter and increases to 15 km in summer.
In and , Monte Carlo techniques are used to
forecast the drift of an object on the ocean surface. They associate the
density of trajectories at their end points to a density of probability and
use them to define a search area, within which the object is likely to be
found. The search area is characterised by a surface centred on the ensemble
mean and which size increases with the ensemble spread. The same methodology
is followed here for forecasting the location of an object on drifting sea
ice. In the context of rescue operations, the search area should be large
enough to contain the actual position of the object but not excessively
large, so as to keep the rescue operation time period and resources affordable and
efficient. The forecast system should therefore ideally yield a high
probability of finding the object in the search area, while keeping at the same
time the search area as small as possible for the cost efficiency of the
rescue procedure.
The probability of finding a drifting object inside the search area is referred
to as the probability of containment (POC) and computed by counting
the objects falling within the search area divided by the total number of
objects. The POC may be interpreted as the ratio of the size of the search
area to the square forecast error. Thus, a small forecast error compared to
the search area leads to a strong POC; conversely, a small search area
(ensemble spread) compared to the forecast error leads to a poor POC.
In order to evaluate the probabilistic forecast capabilities of neXtSIM and
the more classical FD model, the context of a search and rescue operation
is adopted. We assume that an IABP buoy has been lost for 10 days: its
initial position, x0 (see Fig. ), is assumed
to be its last known position. The search area is then defined as the
smallest ellipse centred on the ensemble mean position, B(t),
encompassing all simulated members of the ensemble at time t. The main axes
of the ellipse, a∥ and a⟂, are aligned, respectively, with
the parallel and perpendicular directions from the initial position, as
defined in Sect. and illustrated in Fig. .
Similarly to Eq. () an anisotropy ratio R=a∥/a⟂ can
be defined: R can be large due to the sea ice rheology. A search area
defined in this way increases with the ensemble spread and contains 100 %
of the ensemble members.
Due to a lack of related literature for search and rescue in sea ice, we consider the
values of open-ocean search areas and POC found in and
as references. These are, respectively, of the order of
1000 km2 and 0.5 after 2 days of drift in the North Atlantic. We do not
expect, however, a direct correspondence of these values to those of this
section. First, the sea ice is solid and held together by the ice rheology, in
particular in high concentration areas, so that the ensemble spread is
expected to be smaller than in the open ocean. Second, the currents in the
North Atlantic are generally stronger than in the Arctic Ocean. Finally, the
search areas may be more complex than just an ellipse; it may well be a set
of disjoint areas, each one with an associated different POC
e.g..
Figure shows the evolution of the ellipse areas,
averaged over all IABP buoys. The increase is nearly linear for both model
configurations and seasons. After 2 days of drift in neXtSIM, the
area does not exceed 100 km2 in summer and not even half as much in
winter. The area is larger in FD, and there is very little difference from
winter to summer. The area for the FD is around 200 km2 after 2 days and
it reaches 500 km2 after 5 days. The search area in FD is about 7 times
larger than in neXtSIM in the winter and 2.5 times larger in the
summer. Therefore, even if the forecast errors are smaller in
neXtSIM than in FD, its shrunken search areas lead to a smaller POC
for neXtSIM than for the FD model (not shown).
On Fig. , we show the spread–error relationship for both
periods and both models. The curves represent the spatial mean of the
forecast error. Overall, the curves are above the black line, indicating that
the forecast error is larger than the spread for both models. The
probabilistic forecasts from both model during both period are therefore
too optimistic: they underestimate the uncertainties of their
forecast. However, it is interesting to note two properties of
neXtSIM. First, for spreads larger than 4 km, the forecast error
from neXtSIM becomes independent from the spread, unlike FD, the
errors of which grow monotonically. Second, for large spreads (greater than 3.5 km in
winter and 6 km in summer) the curves from neXtSIM are consistently below
those from FD and getting closer to the spread. Contrarily to the previous
results, the FD and neXtSIM models behave very differently in the
summer.
Illustration of the forecast error and the anisotropic search area.
Blue dots represents the position of one member, while the barycentre of the
ensemble (its mean) is B(t). The observation O(t) is in green
and the forecast error is defined in Eq. (). See text for
definitions of the search ellipse and anisotropy ratio.
The small values of the spread correspond to shorter forecast lead times (see
Fig. ) and these are the times when the
neXtSIM model is still heavily influenced by its initial conditions
of damage, as previously noted on the anisotropy ratio
(Fig. ). As the damage is irrelevant to the FD model, the
initial error grows slower initially, but keeps growing while the rheology
maintains the errors closer to the spread in neXtSIM.
It should be no surprise that the two models underestimate the errors since
this is a common behaviour of probabilistic forecast systems, but the
differences of the shape of the spread–error relationships indicate that the
two models underestimate the errors for different reasons: the neXtSIM
ensemble is lacking spread during the initial times of the forecast but the
asymptotic convergence of the spread to the errors tends to
be the cause of the constitution of the
initial ensemble.
If one had considered the more linear spread–error relationship in the FD
model alone, it would have been tempting to increase the variance of wind
perturbation errors until a perfect match of the spread to the errors was
obtained, but this would have over-tuned the variance of the wind and masked
that the FD model suffers from unresolved physics.
Time evolution of the averaged ellipse areas for neXtSIM (solid lines) and FD (dashed lines) in winter (blue) and in summer (red).
Spread–error relationship for 12 h averages. The curves are based
on the spatial mean of the forecast error shown as a function of the spread.
Relevance for search and rescue operations
Whether a prediction model is too optimistic or too pessimistic may be
equally problematic in view of search and rescue operations. In practice, the
resources available for search and rescue operations are limited and only a
given area can be covered, although the shape of the area (centre and
eccentricity in the case of an ellipse) may not influence the cost
significantly. Thus, rather than looking at the size of the search area as
estimated from the ensemble model prediction, the search-and-rescue operation
can be posed as follows: for a given area to be searched, which model
forecast gives the ellipse that is most likely to contain the object?
Time evolution of POC according to the search area for
neXtSIM (solid lines) and FD (dashed lines) in winter (a)
and in summer (b) for different time horizons.
The ensemble forecast provides the expected position, B(t), and the
anisotropy, R(t), of the ellipse as defined previously, but the ellipse area
is left free to grow homothetically from 1 to 3000 km2. The POC increases
then accordingly as the observed buoy position is more and more likely to
fall within the ellipse. The dependency between the search area and the
associated POC defines the so-called selectivity curve, which makes
a straightforward model comparison possible: the higher the selectivity
curve, the better the model's ability to locate the searched object. The
selectivity curves also allow an immediate evaluation of the rate at which
predictive skill is lost as a function of time.
For each time t0+Δt, with Δt∈12,24,36,48,…,10×24 h, we compute the POC corresponding to search
areas ranging from 1 to 3000 km2 for both models and seasons. Results
from neXtSIM (solid lines) and FD (dashed lines) are shown at t0+1, 2, 3 and 7 days in Fig. . In winter, for a given area,
the POCs from neXtSIM are almost always above those from FD except
in two cases: at t0+1 day for search areas larger than 100 km2 and at
t0+2 days for search areas larger than 500 km2. If we neglect the
anisotropy for these cases (i.e. consider circular search areas), the POCs
from neXtSIM become larger than FD. This indicates that the strong
anisotropy in neXtSIM is more a disadvantage for small time horizon
and large search areas in this experiment. Otherwise for smaller areas,
larger time horizons or in summer, considering circular or ellipsoidal search
areas makes no difference (not shown). As long as the drift is longer than
3 days, the selectivity curves of neXtSIM are systematically above
FD. However, in summer, for any time horizon and any POC, the results are
very similar with a faint advantage to neXtSIM. When comparing to
POCs of ellipses centred on forecasts from a deterministic neXtSIM
run (not shown), the results are identical in winter and poorer with the
deterministic run in the summer.
For both periods and both models, all curves exhibit a sigmoid shape with an
inflexion point, the position of which depends on the time horizon (higher POC and
larger search areas for longer drift duration). For a 7-day drift in winter
and a POC equal to 0.5, the area is around 300 km2 with
neXtSIM, while it reaches 1000 km2 in FD. In the summer, a
larger area is necessary to obtain the same POC for both models. For a given
search area, the gap between the POCs from neXtSIM and FD seems
independent of the drift duration in summer, whereas in winter it increases
with the time prediction horizon. It is interesting to note the lowermost
value of the POC for small areas in winter, which remains above 0.1 for
neXtSIM. This could be a consequence of the capability of
neXtSIM to simulate immobile ice, while the FD ice is always in
motion with the winds and currents.
Time evolution of the POC difference between neXtSIM and FD
for a search area equal to 50 km2 in winter (blue) and equal to
175 km2 in summer (red).
How do the different models perform for different forecast time (i.e. drift
duration)? To answer this question, we study the time evolution of the
difference between the neXtSIM and FD POCs: when this difference is
positive (negative), neXtSIM (FD) is outperforming FD (neXtSIM).
The POC for both models is evaluated for a fixed search area – a vertical
section across the selectivity curves – equal to 50 km2 in winter and
175 km2 in summer, and the results are shown in Fig. . The
chosen values of the search areas, 50 and 175 km2, correspond to the mean
ellipse areas based on the ensemble spread from neXtSIM after
3 days, averaged over the IABP dataset (see Fig. ),
respectively, in winter and in summer. Figure reveals that,
after 2 days in the winter, the POC of neXtSIM is larger by about
0.2 than the POC of FD; most remarkably, such a substantially improved skill
is then maintained almost stably up to the last day of simulation (10 days).
During the summer, the POC of neXtSIM is also generally higher than
the one of FD, but the difference is half of the one observed in the winter.
Furthermore, after the third day, the difference between the two models
decreases to vanish completely between days 8 and 9. The fact that most of the
superiority of neXtSIM is found during winter is logical and should
be no surprise given that during in the summer the ice mechanics of the two
models are similar.
The negative values for lead time shorter than 1 day in winter are again
likely caused by the initialisation of the neXtSIM ensemble and
another reason to constrain its initial anisotropy to observations.
Discussions and conclusions
The ensemble model sensitivity experiment carried out with neXtSIM
and with an FD model reveals the prominent role of the rheology, which marks
the key difference between the two models. On average over the whole Arctic
neXtSIM is less sensitive to the wind perturbations than the FD,
although large seasonal and regional differences are observed. This is
exemplified by the imprint of the ice thickness field in the ensemble spread
from neXtSIM and the much smaller sensitivity of neXtSIM in
winter than summer, in contrast to the FD model (Figs. and
). Both aspects point clearly to the role of the rheology, which
accounts for the ice thickness and compactness. This behaviour should be
expected to hold also for other sea ice rheologies than the EB.
The two models have been tuned on different observations of ice, seen as in
free drift by each model, so that the different performances originate by the
differences in the resolved model physics at their best performance. The
diffusion regimes of neXtSIM and FD are very different in winter
(Fig. ): the offset between the curves indicating differences
of sensitivity and the slopes indicating different rates of increase and
thus sea ice diffusivity. The expected differences between summer and winter
are only represented when the rheology is turned on.
Due to the dispersive properties of the sea ice, the shape of the ensemble of
simulated buoys positions is generally anisotropic. Such anisotropy is a
signature of the underlying mechanism that drives the dispersion of the
members, which is the shear deformation of the ice cover along active
faults/fractures in the ice. This mechanism is missing in the absence of
rheology (like in the FD model) and represents a clear strength in principle
for the EB rheology in neXtSIM, although with the
present ensemble initialisation it did not prove to be a practical
advantage. Other rheological models, such as the elasto-viscous plastic
model, also present some degree of anisotropy , although
the two models have not been compared in the same conditions.
The performance of the two models differs significantly when forecasting the
trajectories of IABP buoys. The ensemble mean position errors are larger in
the summer (5 km after 1 day and 12.5 km after 3 days of drift for
neXtSIM, about 16 % below the FD results) and consistent with the
values reported by (RMSEs of 6.3 and 14 km,
respectively, but using different time periods). The corresponding errors are
smaller in winter, especially for neXtSIM (25 % smaller than FD),
and down to 4 km for a 1-day drift and 7.5 km for a 3-day drift. These
values seem competitive compared to the year-round average RMSE of
5.1 km per day in the TOPAZ4 reanalysis , even though the ice
drift measurements are assimilated in TOPAZ4 . The RMSEs of the free-drift model in also seem to be higher
than 5 km per day.
The model sensitivity to wind perturbations has been evaluated, yielding (for
10 days of drift) a spread from 5 to 10 km, for winter and summer, respectively,
but this is smaller than the corresponding errors (15 km from the barycentre
to the observations in Fig. ). Still, since the diffusion
regime is respected (at least in the winter), we are confident that the
spread simulated by the model is physically consistent. However, other methods for
perturbing the winds should be tested to remove the super-diffusive behaviour
in summer.
To further improve the spread–error relationship, alternative sources of
errors should be considered such as model initial conditions
and forcings (ice thickness, concentrations, damage, ocean currents). Since
the errors are increasing faster in the first days of the simulations, the
more likely source of local and short-term errors lies in the position and
orientation of the sea ice fracture network, which is left unconstrained in
any of the experiments presented here.
Although we would expect an increase of the ensemble spread if the ice
thickness, concentrations and ocean currents had been taken into account in
the ensemble initialisation, we do not believe it would lead to a much larger
spread, especially in the winter. We suggest instead that, in the perspective
of efficient sea ice forecasting, major efforts should be directed toward
assimilating the observed fractures (as of satellite images). The
assimilation of fracture (as objects rather than quantitative observations)
represents a priori a challenging avenue in terms of data assimilation, which
traditionally deals with quantitative scalar or vector observations; however,
we envision that the damage variable in neXtSIM, showing localised
features, can be constrained quantitatively to deformation rates as derived
from observed high-resolution ice motions and serve as “object
assimilation”.
In spite of the biases, the selectivity curves indicate that a probabilistic
forecast using neXtSIM is largely more skilful than the traditional
free-drift model, and it has the larger potential for practical use in search
and rescue operations on sea ice. Since the Arctic is not easily accessible,
forecast horizons of 5 to 10 days are probably the most relevant for
logistical reasons. On those timescales, the differences of POC shown in
Fig. indicate that the free-drift model gives a poorer
information in winter because of the biases in the central forecast location
and the lack of anisotropy, while in the summer the use of an EB
rheology is only marginally advantageous. The comparison of deterministic
versus probabilistic forecast gives, as expected, an advantage to the average
of the probabilistic forecast, although it is rather small and surprisingly
more important in the summer, although the model non-linearities are stronger
in the winter.
The physical consistency of the ensemble sensitivities is a necessary
condition to the success of ensemble-based data assimilation methods
, which constitutes one follow-up research
direction the authors are currently considering. Combining the modelling and
physical novelty of neXtSIM with modern observations of the Arctic
is seen as a major asset for forecast and reanalysis applications.
Besides the potential use of observations of fractures, as mentioned above,
which is indeed another unique advantage of models such as neXtSIM,
ice drift data are also crucial. Observations of ice drift are still seldom
used for data assimilation, and when it is the case, the success is limited
by the lack of sensitivity of the sea ice model see
e.g.. Nevertheless, the main fundamental issue related to the
use of data assimilation, and particularly ensemble-based methods, is related
to
the nature of the Lagrangian mesh of neXtSIM, which also includes the
possibility of re-meshing . This feature, while
essential to the skill of the model in describing the mechanics of the sea
ice with great details, represents a challenge in developing compatible data
assimilation schemes, as the dimension of the state space can change over
time when these re-meshing occur. This problem has recently attracted
attention in the data assimilation research community see
e.g. and it is also a
main area of ongoing investigation of the authors, following the present
study.