View
235
Download
5
Embed Size (px)
Citation preview
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)