Greve and Blatter (2009), Dynamics of Ice Sheets and Glaciers: Errata

Greve and Blatter (2009), Dynamics of Ice Sheets and Glaciers: Errata We have tried to do a thorough job of finding and correcting errors before submitting the manuscript. Nevertheless, the printed edition needs some corrections. We will update this document occasionally as problems are identified. – Ralf Greve, Heinz Blatter Last update: November 20, 2015 Chapter 3 •Page 22, Eq. (3.23): In the last term of the second line, the differential operator d/dtshould be ∂/∂t. So the correct form of the equation is dψ dt=d dtψ(x(X, t), t) =∂ψ(x, t) ∂t + grad ψ(x, t)·∂x(X, t) ∂t =∂ψ ∂t + (grad ψ)·v. Chapter 5 •Page 67, Eqs. (5.26), (5.27) etc., Page 69, Eqs. (5.34), (5.35) etc.: Confusing double use of the symbol Nb. In Eqs. (5.26), (5.27) etc. it denotes the norm of the gradient of the function Fb(implicit representation of the ice base), whereas in Eqs. (5.34), (5.35) etc. it means the basal normal stress. •Page 93, Eqs. (5.132)5−7: All subscripts “η” should be subscripts “ϕ”. 1 •Page 96, Eq. (5.144): The two instances of Hshould be Hn+1. So the correct form is C=                              2(ρg)n ∂h ∂ξ  n−1 Hn+1 Zζ 0A(T0) (1 −ζ0)ndζ0,if Tb< Tm, Cb(ρgH)p−q ∂h ∂ξ  p−1 +2(ρg)n ∂h ∂ξ  n−1 Hn+1 Zζ 0A(T0) (1 −ζ0)ndζ0,if Tb=Tm. Chapter 6 •Page 117, second line above Eq. (6.32): “(Morland 1987)” should be “(Morland 1987, MacAyeal 1989)”. •Page 121, Eqs. (6.46) and (6.47): In the second (y-component) of each of these equations, txx|z=band tyy|z=bshould be tD xx|z=band tD yy|z=b, respectively. So the correct form of Eq. (6.46) is 2tD xx|z=b ∂b ∂x +tD yy|z=b ∂b ∂x −ρgH ∂b ∂x +txy|z=b ∂b ∂y −txz|z=b =−ρswg(zsl −b)∂b ∂x , 2tD yy|z=b ∂b ∂y +tD xx|z=b ∂b ∂y −ρgH ∂b ∂y +txy|z=b ∂b ∂x −tyz|z=b =−ρswg(zsl −b)∂b ∂y , and the correct form of Eq. (6.47) is 2tD xx|z=b ∂b ∂x +tD yy|z=b ∂b ∂x +txy|z=b ∂b ∂y −txz|z=b= 0 , 2tD yy|z=b ∂b ∂y +tD xx|z=b ∂b ∂y +txy|z=b ∂b ∂x −tyz|z=b= 0 . •Page 126, third line below the heading “6.4 Ice Shelf Ramp”: “... problem:” should be “... problem (Weis 2001, slightly modified and extended):”. 2 Chapter 7 •Page 154, caption of Fig. 7.5: The passage “dashed lines for Λ = 0.25, solid lines for Λ = 1, dash-dotted lines for Λ = 100” should be “dash-dotted lines for Λ = 0.25, solid lines for Λ = 1, dashed lines for Λ = 10”. •Page 159, Eq. (7.50): The third column (under η) of the dimensional matrix contains a sign error. The correct form is . . . η . . . . . . −1. . . . . . −1. . . . . . 1. . . •Page 161, third line above Eq. (7.62): “where pis” should be “where pbis”. •Page 161, Eq. (7.62): The right hand side peff pshould be peff pb . •Page 181, second line above Eq. (7.120): “δ(∆¯u)” should be “δ(∆¯vx)”. Chapter 8 •Page 187, first line of subsection 8.2: “From seismic studies” should be “From seismological studies”. •Page 191, Eq. (8.12): The arguments of Hshould be ˇx, ˇyrather than x, y. So the correct form is wss(x, y) = ZAice ρgH(ˇx, ˇy)G(x, ˇx, y, ˇy) dˇxdˇy . Chapter 9 •Page 204, second paragraph, second/third line: “Placidi et al. (2009)” should be “Placidi et al. (2010)”. •Page 205, Fig. 9.1: In some early copies of the book, this figure was rendered incorrectly. It should appear as follows: 3 9.1 Induced Anisotropy 205 Fig. 9.1. Decomposition of the stress vector into a part normal and a part tangential to the basal plane of the ice crystallite. assume that only the stress component Sttangential to the basal plane contributes to its shear deformation, while the component normal to the basal plane has no effect. According to Fig. 9.1, the decomposition of the stress vector reads t·n=(n·t·n)n+Stet,(9.1) where etdenotes the tangential unit vector. Inserting the decomposition (3.133) of the stress tensor tfor incompressible fluids readily eliminates the pressure pand leaves tD·n=(n·tD·n)n+Stet.(9.2) As mentioned above, deformation of the crystallite in the polycrystalline aggregate is attributed to the tangential component Stonly. Since we aim at a theory which describes the effects of anisotropy by a scalar, anisotropic flow enhancement factor, we define the scalar invariant S2 t=(tD·n)2−(n·tD·n)2.(9.3) This quantity has the unit of a stress squared, and a natural way to nondimensionalise it is by the square of the effective stress σe[Eq. (4.9)], which is also a scalar invariant. Thus, we introduce the crystallite deformability,which is loaded by the stress t,as A(n)=5 2 S2 t(n) σ2 e =5 S2 t(n) tr (tD)2.(9.4) The factor 5/2 has been introduced merely for reasons of convenience, as it will become clear below. •Page 207, Fig. 9.2: In some early copies of the book, this figure was rendered incorrectly. It should appear as follows: 9.1 Induced Anisotropy 207 Fig. 9.2. Uniaxial compression on single maximum (UC/SM) and simple shear on single maximum (SS/SM) for a small sample of polycrystalline ice. Stresses are indicated as black arrows, and the single maximum fabric is marked by the dark-grey arrows within the ice sample. in spherical coordinates, integrating over the zenith angle θand the azimuth angle φ; best to be done with a computer algebra tool] yields a deformability of A= 1 for the isotropic polycrystal. For that reason, the factor 5/2 has been introduced in Eqs. (9.4) and (9.9). The CAFFE flow law for anisotropic polar ice can now be formulated. Essentially, we keep the form of Glen’s flow law (4.16), but with a scalar, anisotropic enhancement factor E(A), D=E(A)A(T)σn−1 etD.(9.10) The function E(A) is supposed to be strictly increasing with the deformability A, and has the fixed points E(0) = Emin (uniaxial compression on single maximum), E(1) = 1 (arbitrary stress on isotropic fabric), E(5 2)=Emax (simple shear on single maximum). (9.11) The “hard” case (9.11)1and the “soft” case (9.11)3are illustrated in Fig. 9.2. Note also that the deformability cannot take values larger than A=5/2. As for the detailed functional form of the anisotropic enhancement factor, experimental data suggest that the enhancement factor depends on the “Schmid factor” (which can be identified with the shear stress in the basal plane, St, of the CAFFE model) to the fourth power (Azuma 1995, Miyamoto 1999). Since the polycrystal deformability Acontains a dependency of S2 t[see Eq. (9.9)], it is reasonable to assume a dependency of Eon A2. However, this does not allow Eq. (9.11) for arbitrary choices of the parameters Emin and Emax to be fulfilled. Hence the function E(A) is chosen to depend on A2 in the interval [1,5 2] only, and for the interval [0,1] a dependency on Atis introduced. The exponent tis adjusted such that the function is continuously differentiable at A= 1. This yields •Page 246, Eq. (9.159), Page 247, Fig. 9.23, Page 248, Eqs. (9.165), (9.166), (9.169)3,4: All instances of γshould be α(inclination angle). •Page 251, caption of Fig. 9.24: The caption should be “... H= 200 m, α= 4◦,Ts=−3◦C, a⊥ s=a⊥ m= 0.2 m a−1,vbx= 5 m a−1,n= 3, A= 5.3×10−24 s−1Pa−3,ρ= 910 kg m−3,κ= 2.1 W m−1K−1,c= 2009 J kg−1K−1, L= 3.35 ×105J kg−1and g= 9.81 m s−2.” •Page 252, caption of Fig. 9.25: The caption should be “... H= 200 m, α= 4◦,Ts=−10◦C, a⊥ s=a⊥ m=−0.2 m a−1,vbx= 5 m a−1,n= 3, A= 5.3×10−24 s−1Pa−3,ρ= 910 kg m−3,κ= 2.1 W m−1K−1,c= 2009 J kg−1K−1, L= 3.35 ×105J kg−1and g= 9.81 m s−2.” •Pages 255-259, subsection 9.3.8: In this subsection (“Enthalpy Formulation”), it has tacitly been assumed that the specific heat of ice is a constant (not temperature-dependent). References Cited or Recommended •Page 265, reference to Calov and Greve (2006): “http://www.pik-potsdam.de/~{}calov/heino.html ” should be 4 “http://www.pik-potsdam.de/~calov/heino.html ”. •Page 265, reference to Durand, Gagliardini, Zwinger and Le Meur (2009): The list of authors should be “Durand, G., O. Gagliardini, T. Zwinger, E. Le Meur and R. C. A. Hindmarsh”. The bibliographical information should be “Annals of Glaciology,50 (52), 109–114”. •Page 266, reference to Gagliardini, Gillet-Chaulet and Montagnat (2009): The bibliographical information should be “Low Temperature Science,68 (Supplement Issue ‘Physics of Ice Core Records II’, Ed. T. Hondoh), 149–166”. •Page 266, reference to Greve, Placidi and Seddik (2009): The bibliographical information should be “Low Temperature Science,68 (Supplement Issue ‘Physics of Ice Core Records II’, Ed. T. Hondoh), 137–148”. •Page 268, reference to MacAyeal (1989) should be added: “MacAyeal, D. R. 1989. Large-scale ice flow over a viscous basal sediment: theory and application to ice stream B, Antarctica. Journal of Geophysical Research, 94 (B4), 4071–4087”. •Page 270, reference to Placidi, Greve, Seddik and Faria (2009): The reference should be “Placidi, L., R. Greve, H. Seddik and S. H. Faria. 2010. Continuum-mechanical, Anisotropic Flow model for polar ice masses, based on an anisotropic Flow Enhancement factor. Continuum Mechanics and Thermodynamics, 22 (3), 221–237. doi:10.1007/s00161-009-0126-0.” 5