Introduction
In an earlier paper (Reference WeertmanWeertman, 1973[a]), the existence of bottom crevasses in floating ice shelves was proposed on theoretical grounds. Bottom crevasses have subsequently been “seen” with the aid of radar within the Ross Ice Shelf by Reference CloughClough (1974) and within the Larsen Ice Shelf by Reference Swithinbank and HusseinySwithinbank ([c1978]). Bottom crevasses that extend above the water-line of a tabular iceberg are shown in a figure of a paper of Reference Weeks, Mellor and HusseinyWeeks and Mellor ([c1978]).
Because floating ice shelves and the tabular icebergs that break off of them are cold it might be expected that bottom crevasses within them are short-lived features. A bottom crevasse is, of course, filled with water. This water must freeze continuously to the walls of a bottom crevasse within a cold ice mass if there is no appreciable circulation of water into and out of the crevasse. But creep deformation can cause continuous opening of a crevasse. Thus whether a bottom crevasse remains open for an appreciable length of time depends upon the relative importance of two competing processes. The purpose of this short paper is to make an approximate calculation of the rate at which a bottom crevasse closes or opens. Such a calculation should be of interest to the problem of towing tabular icebergs. Bottom crevasses might lead to rapid disintegration of towed tabular icebergs. Virtually no attention was given to bottom crevasses in the papers in the very recently published book on iceberg utilization (Reference HusseinyHusseiny, [c1978]).
Theory
It can be shown (Reference WeertmanWeertman, 1973[a]) that an isolated crevasse which contains no water at the upper surface of an ice mass (a top crevasse) penetrates downwards to a depth L given by
where ρ is the density of ice, g is the gravitational acceleration, and σ is the tensile stress that exists within the ice mass in addition to the hydrostatic stress component. In a floating ice shelf the average value of σ is equal to
where h is the ice thickness and Δρ is the difference in the density between sea-water and the average ice density of the ice shelf. Thus an isolated, water-free top crevasse in an ice shelf has a depth L equal to
For high-density ice the crevasse depth is about 7% of the ice thickness.
It can also be shown (Reference WeertmanWeertman, 1973[a]) that a bottom crevasse in an ice shelf, which of course is filled with water, penetrates upwards to a height that is approximately a factor ρ/Δρ larger than the value of L that is given by Equations (1) and (3). Thus a bottom crevasse has a length L equal to
or a length that is equal to 78% of the total thickness of an ice shelf. A pair of top and bottom crevasses, one above the other, could account for about 85% of the cross section of an ice sheet or tabular iceberg. (The length L that is given by Equation (4) is probably an underestimate of a bottom-crevasse length. It was found under the assumption that a crevasse moves into an infinite half-space. The upper surface of a floating ice shelf will cause the actual value of L to be larger than that given by Equation (4).)
The calculations that lead to the equations just given are made under the assumption that ice is an elastic solid with essentially zero fracture strength. Since the fracture process that occurs during the initial cracking takes place rapidly, ice indeed does act in this short time period as an elastic solid. The correction to the value of L because ice has a finite fracture strength is very small when the value of σ is within several orders of magnitude of the value σ = 0.1 MPa (1 bar). One need only use the experimental values of the critical stress intensity factor KC that are collected from the literature and tabulated by Reference Smith and HusseinySmith ([C1978]) to demonstrate this fact.
The critical value Lc that a pre-existing crack must have before it can grow in a catastrophic fashion into a crevasse under a stress σ is not a small length. The values of Kc far ice that are listed by Reference Smith and HusseinySmith (1978) are of the order of 0. 15 MPa m½. Since σ(πLc)½ = Kc the value of Lc is equal to 0.7 m when σ = 0.1 MPa. This value of σ would exist in an ice shelf 200 m thick. However, the critical crack length is probably considerably smaller than the value just calculated. Reference Johnston and ParkerJohnston and Parker (1957) showed that the fracture strength of ice is reduced by a factor of three in the presence of surface-active agents. Salt water was one of the effective surface-active agents they used in their experiments. Thus the value of Kc should be reduced by a factor of three in the presence of sea-water and the critical crack length Lc will be reduced by one order of magnitude.
Cracks at the bottom of an ice shelf might be formed by the etching by sea-water of grain boundaries within the ice. If sea-water freezes to the bottom, surface cracks could form through brine entrapment. Of course, if processes such as bottom melting prevent the formation of surface cracks, bottom crevasses will not form because they cannot be nucleated.
The crevasse length given by Equations (1), (3), and (4) apply to isolated crevasses. In a field of closely spaced crevasses, ones whose spacing is appreciably smaller than their lengths, the crevasse length is reduced by a factor of 2/π. (When the fracture strength of ice is finite, crevasse spacing must satisfy the stability criterion recently investigated by Reference Nemat-NasserNemat-Nasser and others (1979).)
Freezing shut
Bottom crevasses in a cold ice mass might heal themselves shut after their formation through the freezing of water. The average displacement λ with which an air-filled top crevasse or a water-filled bottom crevasse is opened elastically is approximately equal to (Reference WeertmanWeertman, 1973[a])
where μ is the shear modulus of ice. This thickness of water must be frozen in order to close a bottom crevasse.
Inserting Equation (2) and (4) into Equation (5) gives
as an estimate of the average displacement within a bottom crevasse immediately after its formation. For h = 200 m and Δρ/ρ ≈ 1/10 with μ = 3 GPa the value of λ is λ ≈ 5 mm. Thus only a small amount of water need be frozen to close the crevasse.
In a time period equal to t cold ice out to a distance r on either side of a crevasse is warmed to a temperature close to the melting point where r is given by
where D is the thermal diffusion coefficient. (Equation (7) is the usual estimate of the diffusion distance in diffusion problems.) If Δθ is the change in temperature of the ice within the distance r, then for a unit area of crevasse an amount of heat equal to 2C Δθr is absorbed by the ice and removed from the sea-water. The rate of closing of the crevasse, –dλ/dt, is thus given by
where H is the heat of fusion of ice and C is the specific heat capacity of ice.
The crack displacement λ is found by integrating Equation (8) and is equal to
where
The crevasse closure time is found by setting λ = 0 in Equation (9). (The actual displacement on crevasse opening varies from a value equal to zero at the crevasse tip to a maximum displacement at the bottom surface. Thus, if the ice-shelf temperature is a constant, crevasse closure occurs first near the tip and last at the bottom surface.) The average time for a crevasse with a 5 mm opening displacement to freeze shut is about 2 min when Δθ = 15 K (using D = 1.5 × 10–6 m2 s–1, C = 2 MJ m–3 K–1, and H = 0.34 GJ m–3). Thus bottom crevasses should freeze shut very quickly if their opening displacement is only an elastic one.
Creeping open
Immediately after a crevasse is created, its opening displacement is primarily an elastic one. But at later times the opening displacement can be the result of creep deformation. The creep-deformation displacement can be orders of magnitude larger than the elastic displacement.
If ice obeyed a Newtonian (that is, linear) creep-deformation equation the crevasse displacement velocity dλ/dt would be very simple to find. The creep problem is formally the same as the elastic problem in this situation. It is only necessary to substitute the term dλ/dt for λ and the viscosity η for the shear modulus μ in the equations. Thus Equation (5) is replaced by the equation
and Equation (6) by
Equation (9) becomes
According to Equation (12), the crevasse never freezes shut if the following inequality is satisfied
where
But the creep law of ice is non-Newtonian. For a uniaxial tension test it is given by the power-law creep equation (except at very high stresses and very low stresses)
where έ is the creep-rate, έ0 is the creep-rate at a reference temperature θ 0 and reference stress σ0, R is the gas content, Q is the creep activation energy, and n ≈ 3. At any stress level σ an effective viscosity η can be defined by the equation
Thus
where
The stress (that is, the non-hydrostatic stress) around a crevasse will range in value from a magnitude of that of σ away from the crevasse tip to very high values right at the crevasse tip. Thus the effective viscosity will take on a wide range of values near a crevasse. A similar situation exists for a crack in a solid that is elastic until a yield stress is exceeded. Above the yield stress the material deforms plastically. In this situation if σ does not exceed the yield stress, intense non-linear deformation occurs only very close to the crack tip. The elastic displacement of the crack faces is almost the same as for a crack in an elastic solid. Thus for a crevasse in a power-law-creep material it is reasonable to use an effective viscosity where the effective viscosity is calculated using the stress level σ. The crack-opening displacement velocity calculated using this particular effective viscosity will be approximately equal to but somewhat smaller than the actual displacement velocity.
At a temperature of – 10°C the creep-rate of ice under a uniaxial stress of 0.1 MPa is about 2 × 10–10 s–1 (see fig. 4 in Reference Weertman, Whalley, Whalley, Jones and GoldWeertman, 1973[B]). If this stress and this temperature are used as reference ones then η 0 = 2.5 × 1014 Pa s.
By combining Equations (2), (13), and (16) the following relationship is found for the condition required to have a bottom crevasse remain open
For n = 3, θ = –10°C, and Δθ = 10 K, Equation (17) predicts that a bottom crevasse in an ice shelf or tabular iceberg that is hicker than about h ≥ 400 m will remain open. (Of course, bottom crevasses in thinner ice shelves can remain open if there is sufficient circulation of water whose temperature is above the melting point into and out of the crevasses. Equation (17) is found under the assumption that such circulation is negligible.)
Partial check
One partial check can be made of the equation for crevasse opening velocity. Reference MeierMeier (1958) measured the velocity with which a water free crevasse opened up in blue ice at the top surface of the Greenland ice sheet. He found that a 25 m deep crevasse opened up at a rate of 1 to 2 mm d–1. The crevasse was in ice of a temperature of – 1°C to – 6°C. The stress σ was not measured but the crevasse was in ice that evidently was extending at a strain-rate of the order of 0.01 year–1. For a temperature of –3°C this strain-rate corresponds to a stress of about 0.084 MPa (using Q = 63 kJ mol–1). (A crevasse depth of L = 25 m according to Equation (1) requires a stress of 0.16 MPa, a value in approximate agreement with this estimate.) For this stress (0.084 MPa) and temperature the effective viscosity η is equal to η = 1.8 × 1014 Pa s. Inserting the values η = 1.8 × 1014 Pa s, σ = 0.084 MPa, and L = 25 m into Equation (10) gives the result that dλ/dt = 1.2 × 10−8 m s−1 = 1 mm d−1, a result that is in reasonable agreement with the measured opening velocity.
Conclusion
The approximate analysis given in this paper shows that bottom crevasses, once formed, are likely to freeze shut in all but the thickest cold ice shelves and cold tabular icebergs.
