The Magic of Ice

Dorthe Dahl-Jensen

Each crystal has an orientation thatcan be described by the c-axis

All c-axis can be plotted on a half-sphere

Seen from above each c-axis representsa dot on the circleThis is a Schmidt diagram.

GRIP

Dye3

NGRIP

GRIP FabricThorsteinsson, 1997, JGRVol 102, No c12

Example of crystal size change over a climatictransformation from Dome C, Antarctica

Duval and Lorius, 1980, ESPL, Vol 48

Isotropic Flow Law:

Glens Flow Lawij = Ase

n-1sij (15)

trace(s2) = 2se2 = (sx

2+sy2+sz

2+2(sxy2+sxz

2+syz2))

The strain rate is thus not only linearly related to the stress deviator in the given direction, but also to the quadratic sum of all the stress deviators acting on the material. It is a non-linear equation.

The flow law exponent n is normally set to 3. The flow law parameter A is a function of temperature (the Arrhenius Equation)

A = Aoexp(-Q/RT)

T is the temperature in Kelvin R is the gas constant (8.314 Jmol-1K-1)Q is the activation energy for creep (60 kJ/mol)

Microscopic Models

Sachs modelTaylor-Bishop-Hill modelViscoplastic-self-consistent modelNearest Neigbor model

Schmidt factor:

S = cossin

x’z’g = Ax’z’gn

0 x’z’g 0 g = x’z’g 0 0 0 0 0

εx = Um-1(μσx - 1/2(2μ-λ)σy - 1/2λσz)

εy = Um-1(-1/2(2μ-λ) σx + μσy - 1/2λσz)

εz = Um-1((-1/2λ(σx+σy) + λσz)

εxz = 1/2Um-1ντxz

εyz = 1/2Um-1ντyz

εxy = 1/2Um-1(4μ-λ)τxy)

where U = 1/2(2μ-λ)(σx-σy)2 + 1/2λ((σy-σz)2 + (σz-σx)2) + ν(τyz

2+τxz2) + (4μ-λ)τxy

2

The m in this theory relates to the n in Glen’s law: m=(n+1)/2. For the isotropic case, λ = μ = 1/3ν.

Shear Strain Rates

A simple approximation to describe the flow is to assume that the shearstress is the only stress deviator (also used in the shallow iceapproximation)

xz = ½(u/z + w/x) = Asen-1 xz

se2 = ½(xz

2)

Assume w/x << u/z and xz = - gh/x(h-z)

u/z = 2A xzn = 2A(- gh/x)n (h-z)n

u(z) = ubase+2A(- gh/x)n [(h)n+1 - (h-z)n+1]/(n+1)

The ice flux Q will be

Q=u(z)dz = Hubase + 2A(- gh/x)n [(h)n+2]/(n+2)