The first coupled general circulation models only contained a thermodynamic sea-ice model in which the ice was stationary within the individual grid cells.83,84 However, it was soon realized that such treatment of the very mobile polar sea ice did not produce satisfactory results. For example, because of sea-ice dynamics, the thickest sea ice in the Arctic is not found in those regions that have the lowest annual mean temperature, but in regions of sea-ice convergence along the northern coast of Canada and Greenland. In areas of divergence or shear, sea-ice dynamics causes the formation of open water and, because of the accompanying increase in heat loss, a major increase in sea-ice volume there during winter.50,73 In summer, on the other hand, sea-ice dynamics accelerates the melting of sea ice: the ice pack is then able to drift into warmer water (particularly in the Antarctic), and the ice-albedo feedback is more pronounced if divergence or shear can lead to open-water areas within the ice pack. Sensitivity studies generally find that these latter effects dominate over the increased ice growth in divergent areas during winter, which is why models with sea-ice dynamics usually have a reduced sea-ice volume in the annual mean.50,85 The inclusion of sea-ice dynamics also leads to a decreased sensitivity of the ice pack to climatic change because of negative ice-dynamics related feedbacks.13,86,87
In general, sea-ice motion is described by Newton's second law, written as78,88
This equation states that the material derivative Dv/Dt = ∂v/∂t + v · ∇v of a certain sea-ice mass per unit area mA is described by the sum of the Coriolis force Fcor, the downhill-slope force mAg∇H (where H is sea-surface height), the atmospheric and oceanic drag forces τa and τw, and by internal stresses ∇ · σ, where σ is the second-order internal stress tensor. While calculation of the first two force-terms is generally straightforward (with sea-surface tilt being either provided by the ocean model, or calculated from the resulting geostrophic currents), major problems arise from the external drag forces and the internal stresses, as described in the following.
Atmospheric and Oceanic Drag Forces
The shear stress between the ice's lower or upper surface and the ocean or the atmosphere, respectively, can for neutral stratification be calculated from the law of the wall, which describes the vertical profile of horizontal velocity U(z) in a fluid relative to a surface as41
Here, u∗ is the horizontal friction velocity, κ is von Karman's constant and z0 is the aerodynamic roughness length that describes the level at which the fluid becomes motionless close to the surface. The aerodynamic roughness length is often approximated as one tenth of the actual roughness of the surface. Representing the aerodynamic roughness length as and making use of the fact that in general friction velocity is related to the surface shear stress via τs = u∗u∗/ρ, it follows from Eq. 2 that the atmospheric or oceanic shear stress τs can be expressed as41
Here, c is the drag coefficient for the atmospheric or oceanic flow measured at level z. Equation ?? is today the standard approach for calculating the atmospheric or oceanic shear stress on the ice. An additional turning angle between τs and U is needed to account for the Ekman spiral if this is not resolved by the atmospheric or the oceanic model component, respectively. Usually, both this turning angle and the drag coefficient c are assumed to be constant, even though both depend on the aerodynamic roughness length z0 that varies widely over a natural sea-ice cover. A more realistic representation of the variation of z0, which should in particular be possible in models that employ an ice-thickness distribution, is therefore very desirable.
In addition to the shear stress τs, also the form drag of the ice floes τf is crucial, which in this context describes the lateral stresses of winds and currents against the sides of the floes. A recent study89 found that depending on floe size, sometimes more than 60% of the oceanic stress is transferred via τf. A realistic representation of the form drag crucially depends on a realistic floe-size distribution, which, as described above, would also allow for an improved modeling of lateral versus bottom heat transfer between the ocean and the ice. Such improved representation of floe-size distribution seems therefore very desirable, in particular for modeling sea ice in the marginal ice zone that usually consists of small floes.
In addition to these external stresses, also the internal stresses of sea ice are key for any realistic modeling of sea-ice dynamics. These are usually even more difficult to handle, because of the large variety of length scales involved that range from small pancakes that can easily be compacted under convergence, to ice floes of many tens of kilometers in scale, up to the basin-wide scale on which these stresses act to move the ice around (see Table 1). Relating internal stresses to the resulting strain within the ice pack, as described by the ice's rheology, is therefore a major challenge. Since the topic of the most suitable rheology for a large-scale sea-ice model has been widely discussed in the literature (see for example Refs 90 and 91 for extensive recent discussion), I here only provide some context and briefly discuss some of the related open issues from a large-scale modeling perspective.
Table 1. A Hierarchical Classification of Scales of Sea Ice Dynamics with a Description of the Spatial and Temporal Scales of the Sea Ice Dynamics, the Matching of the Atmospheric Forcing to the Sea Ice and the Associated Scale of the Sea Ice Response (Reprinted with permission from Ref 92. Copyright 2003 Blackwell Munksgaard)
|Category||Spatial scale||Temporal scale||Wind coupling||Scale of sea-ice response |
|Floe||<1 km||Hours||Too small||Within floe|
|Multifloe||2–10 km||3 h–2 days||Too small||Floe-floe bumping and redistribution; ridging|
|Aggregate||10–75 km||1–3 days||Too small||Differential floe motions; nonlocal forcing; compression wave; beginning of plastic behavior|
|Coherent||75–300 km||3–7 days||Best direct coupling of atmospheric scales and sea ice deformation||Motion across a field resulting in features such as arched leads, families of sliplines, and progressional compression|
|Sub-basin||300–700 km||7–30 days||Matches in space, but not in time, therefore have temporal smoothing of spatial attributes||Creation of velocity ‘zones’; discontinuities smoothed|
|Seasonal||>700 km||>30 days||Both spatial and temporal averaging occur||Thickness fluctuations and redistribution of ice |
Early models of sea-ice dynamics were based on a purely viscous rheology, but later abandoned in favor of the more realistic elastic–plastic rheology.93,94 This elastic–plastic rheology describes a material that resists convergence elastically until a critical stress is exceeded that causes the material to deform plastically. In sea ice, the elastic behavior stems from the ice's mechanical strength which allows for some resistance to internal stresses. Once the mechanical strength is insufficient to resist compression, the ice deforms plastically to form ridges. A major drawback of this Lagrangian formulation is the fact that because of the reversibility of the elastic deformation, the strain history must in principle be indefinitely taken track of.
Because of this major numerical challenge, the elastic–plastic models were soon abandoned in favor of the viscous–plastic formulation.78 This Eulerian formulation allows for an explicit, numerically efficient solution for the stress as a function of strain rate, whilst still maintaining most of the more physically justifiable properties of the elastic–plastic rheology. The viscous–plastic formulation has had a tremendous success story, forming the core of most dynamic sea-ice models that exist today. A numerically more efficient implementation of this rheology95 is usually referred to as elastic–viscous–plastic. Here the term ‘elastic’ does not refer to a truly elastic physical property of the ice but rather to a numerical construct making use of elastic waves, which allows for an explicit calculation of the stress tensor.
Despite the fact that some form of either the original viscous-plastic formulation78 or its more efficient elastic–viscous–plastic implementation95 have become the de-facto standard in large scale sea-ice models today, this formulation leads to a number of differences between modeled and observed sea-ice movement. Some of these differences might be related to the specific formulation of the viscous plastic rheology, for example the specific choice of a certain yield curve that describes the viscous versus plastic response of the ice cover to external stresses. Most models use the original elliptical viscous-plastic yield curve.78 Modifications of this curve90,91 can improve some aspects of the simulations, as has been suggested for the use of a yield curve derived from a granular model,96 a modified Coulomb yield curve,97 a curved diamond yield curve,98 or a teardrop-shaped curve.99 Such changes in the shape of the yield curve often improve specific aspects of the simulation, and, depending on application, can be preferable over the elliptical yield curve. For the large scale drift pattern that is of interest for climate model simulations, the modified yield curves offer, however, no clear advantage over the elliptical yield curve.99,100 How this finding might change for models with ever increasing grid resolution is currently not clear.
One particular issue that can be improved by adjusting the specific formulation of the yield curve relates to the modeling of sea ice that is attached to ice shelves or land. Such fast ice might be of importance for slowing down ice shelves and marine glacier fronts,101 and its modeling has hence received some recent attention.91,102 It was found that for a realistic modeling of such ice, the proper formulation of the yield curve is crucial. In addition, the maximum viscosity that was originally introduced to achieve numerical stability78 needs to be increased by about five orders of magnitude.91 How such change affects the large-scale flow of nonfast ice is currently not clear.
Other differences between observed and modeled flow field arise from the calculation of ice velocities in the original viscous-plastic formulation. There, these velocities are calculated from the viscosities as estimated from the previous time step. This can cause an inconsistency between the flow field and the ice's viscosity. In order to limit this inconsistency, originally a predictor–corrector scheme with a single pseudo time step of length 0.5Δt was suggested, where Δt is the model's time step. This scheme, however, causes nonrandom errors that usually exceed 1 cm/s in the resulting flow field. In order to converge to errors that can be neglected in large-scale simulations of sea ice, instead often several hundred iterations are needed .91,103
An issue that has been overcome in most implementations of the viscous–plastic rheology relates to the original pressure formulation, which caused a slow, unphysical ice flow even for subcritical stress. This slow flow was set up whenever ice of different thickness (or concentration) existed within the model. While such behavior exists to some degree in any viscous formulation of sea-ice rheology, it was strongly amplified by the original formulation of internal pressure. This behavior is today in most models avoided by introducing a nonlinear-viscous closure scheme (or ‘replacement pressure’) that causes the pressure to approach zero as the stresses approach zero.104
While these issues all relate to the specific formulation of the viscous–plastic rheology, some more fundamental issues might require the move to another rheology. Our modern sea-ice models have for example difficulties in realistically representing the spatial and temporal distribution of sea-ice deformation, causing consistently lower production of sea ice in areas of sea-ice divergence.105,106 A recent study links such difficulties directly to the viscous–plastic rheology, pointing out that this rheology cannot realistically simulate the large, linear kinematic features that form in regions of high strain rate.107 The authors introduce an elastic–brittle rheology which better reproduces the observed distribution of such features. Note, however, that an earlier study found that a realistic representation of linear kinematic features is also possible with a slight modification of the original viscous–plastic rheology.108 This seems to be an issue open to further investigation. In particular, it still is unclear how important an accurate representation of these features really is for a proper representation of large-scale sea-ice dynamics.
Another example for a possibly fundamental issue within the viscous–plastic rheology relates to the observed velocity distribution of sea ice. A recent study found that none of the models used for the 2007 IPCC report reproduces the observed relationship between sea-ice velocity and sea-ice thickness:109 while buoy data indicate that in reality, thin ice moves faster than thick ice, such clear relationship is not found in the model simulations. Since with few exceptions all these models use some derivative of the original viscous–plastic model of sea ice, this shortcoming might directly be related to some fundamental flaw in this formulation, for example the specific formulation of the dependence of sea-ice strength on ice thickness. Additionally, in models employing an ice-thickness distribution, the original formulation of the viscous plastic model does not allow for different ice categories within a single grid cell to move at different speeds, since ice thickness only enters the equations in relation to the ice's strength. This might be an additional reason for the observed differences between observed and modeled sea-ice velocities.
Viscous–plastic rheologies are in addition found to have difficulties in simulating the statistical length-scale distribution of deformation rates.106 While the mean deformation as obtained from measurements is represented well in large-scale models, the extreme deformation rates that create leads and pressure ridges are underrepresented. Because these features are crucial for the realistic representation of the ice-thickness distribution, this shortcoming of the viscous–plastic rheology might be of relevance on seasonal and longer time scales.
There are also some general challenges in the modeling of sea-ice dynamics that are unrelated to the actual rheology that is being used. For example, most models assume an isotropic sea-ice cover, which can become an unrealistic assumption in the presence of aligned leads.90 In addition, most models are based on a continuum hypotheses, which assumes that within any grid box there are sufficiently many ice floes that only their bulk behavior needs to be modeled. While this assumption is justified for any ice cover with a sufficiently large number of cracks that generally weaken the ice pack, it is not clear a priori that the continuum hypothesis still gives reasonable results for, say, 5 km horizontal resolution. Given the large number of studies that have been performed on climatic time scales with rather coarse resolution, it seems certainly safe to assume that the continuum hypothesis is justified for any scale exceeding 100 km. For smaller scales, however, the interaction of individual ice floes plays a larger and larger role, and models that specifically represent such interactions110,111 are therefore likely to gain importance in future coupled models. Again, the quality of such models would crucially depend on the quality of the modeled floe-size distribution.
Another general aspect relates to the representation of the strength of the ice pack. While this strength depends primarily on ice concentration, the ice-thickness distribution is also of importance. In estimating sea-ice strength from the ice-thickness distribution in multicategory sea-ice models, a physically plausible approach is to estimate the strength of the ice pack by considering the ratio of converted potential energy during ridging and total energy.112,113 Such formulations are usually empirical and studies are still largely lacking that validate the resulting temporal changes in the thickness distribution against measured time series of the large-scale deformation tensor invariants. In higher resolution models (<10 km resolution), a direct implementation of such formulations can lead to numerical instabilities because of the possible very wide-spread creation of ridges with accompanying changes in sea-ice strength.114 While this behavior can be overcome by using a very small time step, a more efficient solution results from modifying the so-called participation functions that describe by how much each individual ice-thickness category participates in any ridging event.114 Additional progress in describing the changes in ice thickness distribution might arise from a recent study that describes the mechanical ice-thickness re-distribution by ridging and rafting as a general stacking process,115 extending an elegant study in which the ice-thickness distribution for thick ice was estimated from a stochastic model.116 While much work has been dedicated to the implementation of the formation of ridges, also their ablation deserves some special attention in coupled models. Probably because of their high porosity, ridges are found to ablate much faster than the surrounding ice.117 However, the impact of such interior melting is currently not represented in large-scale models.