CONSTITUTIVE PROPERTIES OF PACK ICE

It was observed already by Nansen (1902) that the relation between ice drift and wind velocity varies much and he explained this on the basis of the internal friction within the ice;

he stated that internal friction increases with increasing compactness. This reasoning was confirmed by Sverdrup (1928), who tried to formulate the internal friction empirically through a frictional force proportional to ice speed and directed against ice drift.

What we need is a constitutive law which gives the stress tensor I (two-dimensional) within the ice pack as a function of the state of the ice, deformation etc. The equation of motion tells us that the frictional force within the ice is equal to the divergence of 2. It must be remembered that we are here dealing with mesoscale stresses within the ice pack. Locally the stress can be much larger. E.g., the ice stress felt by a ship can exceed the mesoscale stress by two orders of magnitude (Kheisin 1978).

The first physical constitutive law for pack ice was proposed by Laikhtman (1958). He considered the motion of an ensemble of ice floes analogous to that of a turbulent fluid.

With the classical approach to the Reynolds stresses, Laikhtman obtained

1= 2 riei', (4.3)

where å' = i — '/z (trr)I is the strain-rate deviator and rl, the dynamic eddy viscosity coefficient. If the gradient of rj, is neglected, the divergence of I becomes equal to rl, V2 V.

Campbell (1965) could simulate the general features of ice drift in the Arctic Ocean with his numerical model using Laikhtman's frictional force. Doronin (1970) emphasized the importance of compactness and stated that rl, = 1e (X).

A general viscous law was proposed by Glen (1970):

I= 4 (tri) I + 2 n i' , (4.4)

where and rl are the bulk and shear viscosity coefficients, respectively, and they can depend on the mass characteristics of ice and the strain-rate invariants. It should be noted that Eq. (4.4) is the classical viscous law not involved with the Reynolds stresses, but their approximation through (4.3) gives a similar form to the turbulent and viscous shear stresses. The data from SI79 allowed calculation of the Reynolds stresses directly: the components of the covariance tensor of ice velocity were at their largest 10-' — 10-2 m2s-2 and hence the Reynolds stresses were 1 — 10 Nm-', which is definitely too small to give a significant internal friction. The same conclusion has been reached for the Arctic Ocean by Rothrock (1975b). Consequently, the viscosity of ice must be analogous to the classical viscosity and not the eddy viscosity.

Glen's law has been widely applied with constant viscosities to give the internal friction as 0•=4V(V-v)+ YlV'v.

A slightly more general form was used by Campbell & Rasmussen (1972) : rl = constant and

~ = 0 or rl for 0•v > 0 or 0•v < 0, respectively. This treatment of the bulk viscosity is physically supported by the ice pack giving high resistance to compression and small resistance to tension. Estimates given for the linear viscosity coefficients vary much (Table 10). Hibler (1977) developed a non-linear viscous law which gives high resistance to small deformations, thus approximating the plastic behaviour of pack ice.

TABLE 10. Estimates for linear viscosities of pack ice. Unit kg s-1

Bulk viscosity Shear Viscosity Area Reference

— "1012 Central Arctic Campbell (1965)

— 3.108 Kara Sea Doronin (1970)

1010-1012 10b0-1012 Central Arctic Hibler & Tucker (1977)

^10$ ^108 Baltic Sea Leppäranta (1980a)

107_10 t0 107-10b0 Bothnian Bay This work

The theory of particulate media (e.g. Harr 1977) suggests that the viscosities for a material such as pack ice should be sensitive to compactness. Doronin (1970) assumed only a direct proportionality between rl and X, but in the range X ~. 0.8 rl can probably vary through one or two orders of magnitude, and consequently the viscocity gradient would become significant. Hibler (1979) had in his model the proportionality of n, g c exp [— 20(1 — X)], which seems to give the correct magnitude of variability.

There are two additional terms which have sometimes been used in connection with Eq.

(4.4). Nye (1973b) showed that the two-dimensional pack ice stress is actually the three-dimensional Cauchy stress integrated thfough the thickness of ice plus a horizontal hydrostatic load of water on the inclined (due to nonzero thickness gradient) lower surface of ice. However, the latter term can be present only in compact ice (X= I); in the spring

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time, when there is always some open water, the horizontal hydrostatic load integrates to zero in length scales larger than the floe size.

The second term is a hydrostatic pressure within the ice pack (Rothrock 1970). The pressure should result from the equation of state for pack ice, but formulation of such has not yet been successful. Hibler (1979) assumed the pressure to be a function of the compactness and mean thickness only — this can give rise to stresses in an ice cover at rest.

Kheisin & Ivchenko (1976) assumed that the pressure is proportional to the displacement divergence — this necessarily brings a reference configuration into the picture and hence does not seem very realistic. To conclude, even though the idea of the hydrostatic pressure itself is acceptable, its formulation is at present far from satisfactory. The equation of state should include not only mass characteristics but also the spatially fluctuating part of the kinetic energy. The close packing of floes and the importance of their collisions evidently makes the equation of state quite complicated.

Completely different from the viscous approach is the treatment of pack ice as an elastic-plastic medium (Coon et al. 1974, Pritchard 1975). The reasoning here is based on small scale mechanical behaviour and features predicted by the elastic-plastic law agree with observations. However, the viscous and plastic approaches are not contradictory, since a viscous law results from averaging stochastic variations in deformation rates even though the nonaveraged law is plastic (Hibler 1977).