1. Outline of the Problem and the Suggested Solution

The oxygen isotope ratios used to identify the annual layers of the Antarctic and Greenland ice sheets gradually lose their initial contrast by self-diffusion through the solid ice. However, recent work (Reference JnhnscnJohnson and others, 1997) on ice from the GRIP core. Greenland, concludes that the usual processes ol solid-state diffusion, including grain-boundary diffusion, are too slow to produce the observed loss of contrast. The diffusion constant in the -32°C cold Holoccne ice needs to be about 10 times greater than the measured single-crystal diffusion constant at the same temperature (Reference RamseierRamseier, 1967a, b) to explain the data. This note suggests a new process, not previously recognised, that will enhance the rate.

It depends on the presence within the ice of wins of unfrozen water at the three-grain junctions (Nye and Frank. 1973; Reference MaderMader, 1992a, b; Reference Nye, Maeno and HondohNye, 1992:. The veins form a continuous network thai renders the ice slightly permeable to water. They do not freeze when the temperature is lowered because o! two physical effects: one is lowering of the freezing point by the curvature of the ice water interfaces at the triple junctions (the ice is convex towards the water); the other is the lowering of the freezing point by soluble impurities in the vein water. As the temperature is reduced the veins tend to freeze, but this increases the curvature, and so lowers the freezing [joint still further; also, because the ice lattice rejects impurities, the)- remain in the liquid, which becomes more concentrated. The result is that a new equilibrium is found at the lower temperature, with the veins narrower but still open.

To see the effect of the veins on the diffusion of isotope ratios, consider two layers with isotope ratios b — bo + A and <5 = i\, - Δ separated by a distance A/2, where λ is the spacing of the annual layers. The standard diffusion process is driven by the gradient 4Δ/?. However, suppose we conceptually suspend this process and connect the layers by water wins. Isotope diffusion within the veins will be so fast compared with solid-state diffusion that ô within them will reach the mean value 6Q virtually instantaneously ι within hours or days). Thus, the rate-controlling process is now lateral diffusion within the layers towards and away from the veins, and the gradients driving this process are ±Δ/γι, where i'| is half the spacing of the veins, which is approximately the grain-size 6. If v, < A/4, that is b < Λ/2. diffusion by this process could outpace the standard process. On dimensional grounds one would expect the rate for die huerai process to be about (X/2b)~ times the rate for the conventional vertical diffusion process. In other words, the veins might short-circuit the usual vertical diffusion and the rate of approach to uniformity would then be controlled by how close a point is to a vein, rather than by the layer spacing. That is the suggestion.

2. More Detailed Model

T) examine the geometry more carefully, with both processes taking place at once, consider a long vertical cylinder of ice of outer radius nand inner radius r\,. The inner cylindrical hole represents a vein, and on its surface b is maintained constant at b_tl. The radius of the outer cylindrical boundary is taken as n = 6/2, where b is t he gra i n-size. Diffusion of 6 takes place with diffusion constant D in the vertical î direction, and also radially with the boundary condition db/dr = (Ion r = n, according to the equation

(1)

Writing ()(r,z.f) = F(r)Z{z)T{t). and substituting. separates the variables to give

(2)

and

(3)

where the k\ arc constants satisfying k — k_r~ + k_t . Thus, a solution can be written

(4)

with F(r) obeying Equation (3) with boundary conditions F(r₀) = 0, f(n) = 0. With the substitutions k,.r = R. k,!\) — R\). k,r\ =R\. Equation (3) becomes the Bessel equation

(5)

with boundary conditions F(RQ) = 0, F'(Rt) = 0.

The factor sink-z in Equation (4) represents the annual layering. Thus we regard λ\- as prescribed, but Ay ι the lowest frequency that Fits the boundary conditions) remains to be found, although it must be of order l/ri. The question w ill be the relative contributions of fe₂ and k_r to the decay constant k. The general solution of Equation (5) is

(6)

the other arbitrary constant being absorbed into A.

The boundary conditions then imply

(7)

and

(8)

which is equivalent to

(9)

r_o and r₁ are given. At -30°C the expected vein size is about r_o = 1 μm (Reference MaderMader, 1992b). With the grain-si/.e in Holoeene ice observed to be about 3-4 mm (Thorsteinsson and others. 1997), we take r ₁ = 3.5/2 = 1.8 mm.

Although Tu and r_o are given, R_o and R₁ arc not, because k_r is unknown. R_o is given in terms R₁ by R₀ = (r_o/r₁)R₁ . Then, eliminating B between Equations (7) and (9) gives the following implicit equation for R₁ :

(10)

which is solved by iteration to give R₁ = 0.544. It follows that k_r = R₁/r₁ = 302m^-1.

This is to be compared with k_z . For ice 10 000 years old the layer spacing is A = 0.1 m, which gives k_z = 2τ/λ 63 m⁻¹. Thus, k_r, from the lateral diffusion process, makes a larger contribution to the rale constant k = k_T ² + k_z ² than does k_:. from conventional vertical diffusion. k_r increases k by a factor / = {k,,² + fc_; ²)/fc_;'² = 24; that is, the diffusion constant will appear to be 24 times greater than normal. The result depends almost entirely on the relative values of i'\ and Λ, the dependence on the vein size rr, being logarithmic only. For a grain-size of 6 mm instead of 3.5 mm, as taken above, the factor would be 8.7.

Because R₁ is always roughly 0.5 in the range of interest, a useful approximation is to take k_r = 1/6, which gives the simple formula f = 1 + (λ²/4πb²).

Samples taken from the interiors of the individual ice grains ought to give the best contrast.

S.J. Johnsen [personal communication) remarks that the excess diffusion only acts in relatively clean glacier ice like the Holoccnc ice; samples from the dusty, last glacial, ice show no evidence of it. He suggests that the veins may be blocked by dust particles in the latter case. Of course, any blockage would have to act on the diffusion process along the veins, not just on the bulk motion of the water.