Sea ice pressure poses great risk for navigation; it can lead to ship besetting and damages. Contemporary large-scale sea ice forecasting systems
can predict the evolution of sea ice pressure. There is, however, a mismatch between the spatial resolution of these systems (a few kilometres) and the
typical dimensions of ships (a few tens of metres) navigating in ice-covered regions. In this paper, the downscaling of sea ice pressure from the
kilometre-scale to scales relevant for ships is investigated by conducting high-resolution idealized numerical experiments with a viscous-plastic sea ice
model. Results show that sub-grid-scale pressure values can be significantly larger than the large-scale pressure (up to

With the growing shipping activities in the Arctic and surrounding seas, there is a need for user-relevant sea ice forecasts and products at multiple timescales and spatial scales. An important forecast field for navigation is the internal sea ice pressure (simply referred to as pressure for the rest of this paper). In compact ice conditions, high-pressure events can complicate navigation activities and even pose great risk for ship besetting.

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By solving equations for the momentum balance and for the ice thickness distribution, sea ice models are able to predict the evolution of the pressure
field. However, even for high-resolution operational forecasting systems with spatial resolutions of a few kilometres (e.g.

Some researchers have done case studies of compressive or besetting events using large-scale sea ice forecasting systems
(e.g.

There is also a vast literature on the performance of ships navigating in ice-infested waters and on the estimation of ice resistance, that is the
longitudinal forces applied on the ship by the ice (e.g.

In contrast with studies mentioned in the last paragraph, we focus on ship besetting rather than on a ship progressing in an ice-covered region. We
also study the downscaling of sea ice pressure from the kilometre scale to scales relevant for navigation activities (tens of metres). Note that this was briefly
investigated by

This paper is structured as follows. In Sect.

The large-scale sea ice forecasting system solves the sea ice momentum given by

The sea ice pressure is by definition the average of the normal stresses, that is

As the stresses are vertically integrated, the stresses and stress invariants are 2D fields with units of newtons per metre. Because the sea ice
stresses are written as a function of the sea ice velocity (see details below), one also obtains the pressure

Here, we consider a small area of sea ice (the size of a grid cell) to be under compressive stresses. The idea is to apply the large-scale pressure at
the boundaries of this small area and to simulate the sub-grid-scale sea ice pressure (referred to as the small-scale pressure). We assume here that
the ice is neither moving nor deforming (e.g. it is being held against a coast). To further simplify the problem, the wind stress, the water stress, the
advection of momentum and the sea surface tilt term are neglected. We wish to find, inside this small domain, the steady-state solution of

The bulk and shear viscosities are, respectively,

Following

When

We use a replacement method similar to the one presented in

The replacement method is commonly used in sea ice models to prevent unrealistic deformations of the sea ice cover when there is no external forcing.

The McGill sea ice model is used for the numerical experiments. We use revision 333 with some modifications, described below, for specifying stresses at the boundaries.

Considering a domain of a few kilometres by a few kilometres wide (representing a grid cell of a large-scale sea ice forecasting system), the idea is to use the model
at very high resolution for studying the distribution of pressure inside that domain. To do so, the model was modified so that internal stresses can
be specified at the boundaries (instead of the usual Dirichlet condition, i.e.

For the experiments, the domain is a square of dimensions

The momentum equation

The spatial discretization of Eq. (

For some of the numerical experiments, a digitized ship is placed inside the domain. The digitized ship is simply defined as a rigid body by using
land cells. The boundary conditions on the contour of the ship are therefore no outflow and no slip (i.e.

The boundary conditions are imposed the same way on the four sides of the small domain. Hence, to shorten the paper, only the treatment on the west
side of the domain is explained here. The McGill model uses an Arakawa C-grid; the center of the cell is the point for tracers (e.g.

One grid cell on the western boundary of the domain with indices

On the west side of the domain, a normal stress (

On the other hand, the term

For the

Even though

In our simulations,

The McGill model has, over the years, been extensively tested (e.g.

In all the experiments, normal and shear stresses are applied at the four boundaries of the 5

Compared to realistic pan-Arctic simulations, the simplicity of the problem allows one to obtain analytical solutions for specific cases. In a first
validating experiment, the thickness (

Pressure field for

We also verify that we obtain the same results when a lead is present within the physical domain for different spatial resolution (

The effect of the same lead but oriented differently in the domain was also tested. The PDF of the pressure field is exactly the same whether the lead is oriented horizontally (west–east) or vertically (south–north; not shown). The spatial distribution of pressure is qualitatively the same when orienting the lead diagonally. The PDF of pressure for this diagonal lead is similar to the PDF of the vertical and horizontal ones, although we find that the maximum pressure is usually a bit smaller (not shown). This is likely a consequence of the spatial discretization of a finite width lead on a Cartesian grid.

In a last set of experiments for the validation, we also checked that the presence of relatively nearby boundaries does not affect our conclusions. In
the first experiment, with

To limit the number of parameters that can be varied in the numerical experiments, the thickness of the level ice

Pressure (

In a first set of experiments, we conduct idealized experiments to investigate the impact of sea ice features (leads, ridges, etc.) on the small-scale
pressure field and especially on the maximum pressure. These experiments will give us insights and guide us for the second series of experiments with
the idealized ship (see Sect.

Maximum value of the shear stress invariant (

Our results above suggest that only the longest lead needs to be considered for estimating the largest small-scale pressure. For a given

As the whole yield curve scales with the value of

Pressure field with

While the average pressure in the domain is the same (10

Schematic of the coarse-graining procedure. The thickness field is defined at 10

We also investigate the evolution of the small-scale pressure field as a function of resolution. The

All the values of pressure

All the values of

Pressure field at 10

In a second set of experiments, we investigate the small-scale pressure field in the vicinity of a ship in heavy sea ice conditions and under
compressive stresses. Importantly, we estimate the maximum pressure applied on the ship in different idealized experiments. The small-scale pressure
field around a ship 90

Maximum pressure (

A crucial aspect to consider here is the length of the lead behind the ship. Assuming the lead closes at a shorter distance from the ship should
imply smaller pressure values (for the same pressure applied at the boundaries). This is indeed the case as demonstrated by the sensitivity study
shown in Fig.

Maximum pressure (

Figure

We have investigated how sea ice pressure could be downscaled at scales relevant for navigation. The distribution of pressure at small scales is
associated with non-uniform sea ice conditions. The PDF of the small-scale pressure is non-symmetric (it is limited by 0 on one side) and is skewed
toward large values. Our results indicate that what really determines the largest values of pressure is associated with defects, that is long
leads. Because a lead itself is not able to sustain any stress (unless it has refrozen), the load is taken by the ice around the lead with especially
large values of the stresses in the vicinity of the tips. A sensitivity study indicates that the small-scale distribution and maximum pressure are
notably affected by the choice of the shear strength (

Idealized experiments with a digitized ship beset in heavy sea ice conditions show that stress concentration also occurs in the vicinity of the
ship. In fact, our simulations show that the largest pressure applied on the ship is found on both sides at the back of the ship. These results are
different than the ones of

We also argue that the ship itself is responsible for the strong concentration of stress on its side; the lead (or channel) it created by navigating in sea ice causes these large values of the stresses. Moreover, it is found that even a short lead causes pressure values notably larger than the pressure applied at the domain boundaries. The stresses on the ship should decrease as the ice in the lead consolidates (by either thermodynamical growth or closing of the lead). These conclusions highlight the difficulty of providing sub-grid-scale pressure forecasts for navigation applications as the important parameters (i.e. the length of the lead and the thickness of the refrozen ice) are not well constrained.

A significant advantage of our numerical framework is that stresses can be specified at the boundaries. However, it is also important to note its
limitations. First, it can only calculate the pressure field for a ship beset in heavy sea ice conditions; it cannot simulate the sea ice stresses
applied on a ship navigating in ice-infested waters (as in

In our numerical experiments, the digitized ship is simply represented as a rigid body with no outflow and no slip boundary conditions applied on the
contour. A more realistic numerical framework should also involve a better representation of ship–ice interactions. For example, as done by

Although the convergence criterion for the steady-state solution of the velocity field has been reached in all the numerical experiments described in
this paper, it is worth mentioning that this came with tremendous difficulties for the JFNK solver; the non-linear convergence was really slow, and the
solver failed on some occasions to reach the required drop in the Euclidean norm of the residual within the allowed 500 non-linear
iterations.

A few observations were made concerning the numerical stability of our new numerical framework with stresses applied at the boundaries. In this
appendix, we discuss and provide explanations for these limitations.

We have noticed that for a simulation to be numerically stable,

For stability,

As advection and thermodynamics are not considered, the thickness field is constant in time, and we can write

In 1D,

The term on the right can be integrated by parts, that is

For the second term on the right,

The stability therefore depends on the boundary term

With

With

This means that

To ensure numerical stability,

Revision 333 of the McGill sea ice model was modified so that stresses can be prescribed at the boundaries. This code is available
on Zenodo at

JFL and BT developed the downscaling method and the modified boundary conditions. JFL modified the model code and conducted the numerical simulations. JFL, BT and MP analyzed and discussed the results. JFL wrote the manuscript with contributions from BT and MP.

The authors declare that they have no conflict of interest.

We thank P. Blain for his comments and for carefully reading the manuscript. We also thank R. Frederking, H. Heorton and an anonymous reviewer for
their very helpful comments. Finally, we would like to thank
Frédérique Labelle and Bimochan Niraula for developing the Python code for Fig.

This research has been supported by the NSERC (grant no. RGPIN-2018-04838).

This paper was edited by Yevgeny Aksenov and reviewed by Robert Frederking, Harry Heorton and one anonymous referee.