Glacier Dynamics. Glaciers 617. Andy Aschwanden. Geophysical Institute University of Alaska Fairbanks, USA. October 2011

Transcription

1 Glacier Dynamics Glaciers 617 Andy Aschwanden Geophysical Institute University of Alaska Fairbanks, USA October / 81 The tradition of glacier studies that we inherit draws upon two great legacies of the eighteenth and nineteenth centuries: classical physics and romatic enthusiasm for Nature.... It is easy to undervalue the romantic contribution, but in glaciology, it would be a mistake to do so Garry Clark, J. Glaciol., 1987)

2 Course Material lecture is based on lectures by Ed Waddington & Martin Truffer Martin s Continuum Mechanics script book recommendation: Greve and Blatter (2009) Greve, R. and H. Blatter (2009). Dynamics of Ice Sheets and Glaciers. Advances in Geophysical and Enviornmental Mechanics and Mathematics. Springer Verlag. 3/81 Introduction to glacier flow 4/81

3 How does a glacier move? In a nut shell: The ice can deform as a viscous fluid The ice can slide over its substrate Well-described by continuum mechanics Measurements & Observations 5/81 Measurements & Observations 1779 Gravitation theory by de Saussure H. B. de Saussure observes sliding... the weight of the ice might be sufficient to urge it down the slope of the valley, if the sliding motion were aided by the water flowing at the bottom Hugi Block J. Hugi observed that a boulder moved 1315 m downstream between 1827 and 1836 we would interpret this as clear evidence of glacier flow but back then, some people argued that a boulder slides on the glacier surface, the glacier itself is motionless Measurements & Observations 6/81

4 Measurements & Observations Dilatation theory by L. Agassiz glacier ice contains innumerable fissures and capillary tubes during the day, these tubes absorb the water and during the night, the water freezes this distension exerts a force and propels the glacier in the direction of least resistance Measurements & Observations 7/81 Measurements & Observations Viscous flow theory by J. Forbes made his own observations on Mer de Glace, France glacier flows fastest in the center opposes Agassi s theory if the dilatation theory were true then flow would be greatest at sunset and near the glacier margins Measurements & Observations 8/81

5 Measuring surface velocities Measuring angles (with Theodolite) and distances (with Electronic Distance Meter or EDM) from fixed stations on the glacier margins GPS (global positioning system) feature tracking in repeated aerial photography or satellite images Measurements & Observations 9 / 81 Measuring surface velocities interferometric synthetic aperture radar (InSAR) count fringes from a stationary point on bedrock credit: USGS, Joughin et al., 2010 Measurements & Observations 10 / 81

6 Measuring borehole tilt with tiltmeter (inclinometer) Measurements & Observations 11 / 81 Kinematic vs. Dynamics Kinematic description of flow In a steady state, Flow required to transport away upstream accumulation Glacier has to adjust its shape to make this flow happen Rheological properties don t figure in kinematic description Dynamic description of flow ice is a material with certain rheological properties (e.g. viscosity) Flow is determined by stresses applied to it Accumulation/ablation don t figure in dynamic description Measurements & Observations 12 / 81

7 Glacier Flow Chart climate meteorological process mass balance this lecture ice dynamics processes glacier geometry Measurements & Observations 13 / 81 Why is it so hard to predict the future of an ice sheet? It s easy because composed of a single, largely homogenous material viscous flow is governed by the Navier-Stokes equations (19th century physics) move very slowly (turbulence, Coriolis force, and other inertial effects can be ignored) It s so hard because the stress resisting ice flow at the base can vary by orders of magnitude ocean interactions could trigger instabilities Measurements & Observations 14 / 81

8 Forces a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape forces exist inside continuous bodies such as a glacier these forces can cause a glacier to deform Forces, Stress & Strain 15 / 81 What is stress? Stress is the force per unit area acting on a material σ = F A Inside a glacier, stresses are due to weight of the overlying ice (overburden pressure) shape of the glacier surface (pressure gradients) Forces, Stress & Strain 16 / 81

9 Types of stress As a force per unit area, stress has a direction Force can be directed normal to the area Result is pressure if the force is the same on all faces of a cube. Result is normal stress if forces are different on different faces Force can be directed parallel to the area Result is shear stress Forces, Stress & Strain 17 / 81 Magnitudes of stress & forces Again, stress is force / (unit area) frictionless table water bottle in free fall stretching a rubber band Forces, Stress & Strain 18 / 81

10 Pressure in a glacier mass m = ρv ρ =icedensity 900 kg m 3 V = Volume = Area depth = A z So pressure p at depth z is p = m g A = ρ A z g A = ρgz How deep do we have to drill into a glacier before the ice pressure is 1 atmosphere? Forces, Stress & Strain 19 / 81 Depth for 1 atm pressure? z = p ρ g =? So pressure rises by 1 atm for every x meters of depth in a glacier Does ice deform in response to this pressure? Forces, Stress & Strain 20 / 81

11 Shear stress τ Total stress t from ice column: t = ρvg = ρ gh How much of this weight will contribute to shear deformation? shear stress τ = ρ ghsin α normal stress σ = ρ ghcos α Forces, Stress & Strain 21 / 81 Shear stress in a glacier h = 130 m α =5 τ = ρ ghsin α Shear stress at the glacier base, τ b,is 1bar, which is a typical value for basal shear stress under a glacier Forces, Stress & Strain 22 / 81

12 Are glacier thickness and slope related? Suppose a glacier becomes thicker or steeper due to mass imbalance: it flows faster it quickly reduces thickness h or slope α, untilτ b 1 bar again Can we then estimate glacier thickness (z = h) from its slope if we know τ b 1bar? τ b τ = ρ gz sin α h ρ g sin α How are shear stress and shear strain rate (velocity) related? Forces, Stress & Strain 23 / 81 Strain rate γ shear strain γ = 1 2 x z shear strain rate γ = 1/2 = 1 2 γ x/ z t = 1/2 u z Forces, Stress & Strain 24 / 81

13 Material Science Rheology is the study of the flow of complex liquids or the deformation of soft solids. different materials respond differently to applied stresses We are not going through this in great detail. For further information, consult the handout or take Martin Truffers Ice Physics 614 class We need to define a relationship between shear stress and shear strain rate, τ = f ( γ) Material Science 25 / 81 Example: Shear Experiment apply constant shear stress τ τ is the applied shear stress γ is the shear angle γ is the shear rate initial elastic deformation primary creep: shear rate γ decreases with time secondary creep: shear rate remains constant acceleration phase tertiary creep: constant shear rate (but higher) Material Science 26 / 81

14 Example: Shear Experiment τ is the applied shear stress γ is the shear angle γ is the shear rate This suggests that the shear rate is a function of the applied shear stress, temperature T,andpressurep γ = γ (τ,t, p) Material Science 27 / 81 Example: Shear Experiment τ is the applied shear stress γ is the shear angle γ is the shear rate The dynamic viscosity η relates shear stress and shear rate τ = η (T, p, γ ) γ Material Science 28 / 81

15 Constitutive behavior of ice γ = 2 A(T, p) τ n 1 τ A(T, p) = A 0 e Q/(RT ) Arrhenius law A is called rate factor (ice softness) Q and R are the activation energy and the gas constant, respectively n is the exponent of the flow law, usually taken as n = 3 This is known as Glen s flow law or Glen-Steinemann flow law but similar to Norton s flow law for the flow of steel at high temperatures Material Science 29 / 81 Temperature dependence With some emperically-derived values for Q we get: Rate factor A [s Pa ] Temperature T [ C] ice at 0 C is about 1000 softer than ice at -50 C Material Science 30 / 81

16 Thin-film flow u z = 2 A(T, p) τ n 1 τ u h u b = h 0 2 A (ρ g sin α z)n dz = We assume u b = 0, i.e. glacier is frozen to the bed Material Science 31 / 81 Basal sliding Despite decades of intensive research, basal sliding remains one of the grand unsolved problems. We know: sliding is a function of basal water pressure and bed roughness if ice is underlain by till, experiments suggest a plastic or nearly-plastic behavior Beyond the scope of this lecture Material Science 32 / 81

17 Recap so far, we have only considered shear stresses while they are often the most important ones in glacier flow, there are other stresses (e.g. normal stresses) which can be important too we have assumed ice surface and bed surface are parallel let s step back and look at the bigger picture yep, we will need basic calculus and some continuum mechanics Material Science 33 / 81 Field equations for ice flow Material Science 34 / 81

18 Recap so far, we have only considered shear stresses while they are often the most important ones in glacier flow, there are other stresses (e.g. normal stresses) which can be important too we have assumed ice surface and bed surface are parallel let s step back and look at the bigger picture yep, we will need basic calculus and some continuum mechanics Material Science 35 / 81 Continuum Mechanics Continuum mechanics is the application of classical mechanics to continuous media. So What is classical mechanics? What are continous media Material Science 36 / 81

19 Classical Mechanics In classical mechanics, we deal with two type of equations conservation (balance) equations (fundamental to physics) material equations (less fundamental) Balance Laws balance of mass balance of linear momentum balance of angular momentum balance of energy Material Laws / Constitutive Equations / Closures often obtained from experiments and theoretical considerations (such as material objectivity) Material Science 37 / 81 Classical Mechanics Densities classical mechnics has concept of point mass determine velocity (and acceleration) by looking at changes (and rate of changes) in position Kinematics how do forces affect a mass? Dynamics Volume Ω has mass m, then mass m = Ω ρ dv, ρ mass density linear momentum mv = Ω ρvdv, ρv linear momentum density internal energy U = Ω ρudv, ρu energy density (1) Material Science 38 / 81

20 General Balance Laws Imagine a volume Ω enclosed by a boundary Ω G(t) = g(x, t) dv (2) S(t) = Ω Ω s(x, t) dv (3) G, g S, s physical quantity and density supply and density Material Science 39 / 81 General Balance Laws G can only change if if there is a supply S within Ω if there is a flux F of G through the boundary Ω F (t) = (gv + φ(x, t)) n da (4) gv φ(x, t) Ω transport of g with velocity v through Ω flux density (defined later) Material Science 40 / 81

21 General Balance Laws General balance law is: d dt Ω dg dt g(t) dv = = S F (5) Ω s(x, t) dv Ω (gv + φ) n da (6) Material Science 41 / 81 General Balance Laws Assume that G = G(t) thus integral and differential can be exchanged Use Gauss Theorem: (gv + φ) n da = Ω We can then write Ω (gv + φ) dv Ω g t dv = Ω s(x, t) dv Ω (gv + φ) dv (7) Material Science 42 / 81

22 General Balance Laws or We made no assumption about the shape and size of Ω We can make Ω infinitely small g t dg dt = g t = s(t) (gv + φ) (8) + v g = s(t) (v + φ) (9) Material Science 43 / 81 Mass Balance the mass of a material volume cannot change, dm/dt = 0 g = ρ φ = 0 s = 0 mass density flux of g through the boundary supply of g from the above we get ρ + (ρv) =0 (10) t known as the continuity equation We assume ice is incompressible (density does not change), then v = 0 (11) Mass Balance 44 / 81

23 Momentum Balance Newton s Second Law total rate of change of momentum P equals sum of all forces F dp dt = F = ω t n da + internal ω f dv external (12) P and F are vector quantities identify the internal stress vector t n = Tn, see p. 31 T is the Cauchy stress tensor external forces f are, e.g. gravity or the Coriolis force (forces are a source of momentum) Momentum Balance 45 / 81 The Cauchy Stress Tensor We ve already introduced shear stress, but forces can be applied on different faces and in different directions T = t xx t xy t xz t yx t yy t yz t zx t zx t zz We identify τ = t xz For example t ez = t xz t yz t zz is the stress vector of a cut along the xy-plane Momentum Balance 46 / 81

24 Momentum Balance (ρv) t = (ρv v)+ T + f (13) g = ρv φ = T s = f By using the mass balance, we can write momentum density flux of g through the boundary supply of g d (ρv) dt = T + f (14) What s new? shear stress τ replaced with Cauchy stress tensor T Momentum Balance 47 / 81 Momentum Balance For glacier ice, we assume ice is incompressible acceleration term d(ρv) dt can be neglected supply of momentum is through gravity only, f = ρg The momentum balance for glacier flow is T + ρg = 0 (15) Momentum Balance 48 / 81

25 Angular Momentum Balance no, I don t want to do the whole derivation here if you re interested (or bored), go through Equations in Greve and Blatter (2009) here, we just use the result the stress tensor T is symmetric T = T T (16) Angular Momentum Balance 49 / 81 Energy Balance only the total energy (sum of kinetic and internal energy) is a conserved quantity balance of kinetic energy is not an independent statement but a mere consequence of the momentum balance here, we formulate the balance of specific inner energy, u: ρ du dt = q + tr (T D)+ρr (17) g = u φ = q p = tr (T D) s = ρr D = 1/2 v T + v specific internal energy heat flux dissipation power specific radiation power strain rate or velocity gradient tensor Energy Balance 50 / 81

26 Balance Equations mass momentum internal energy dρ dt = ρ v (1) ρ dv dt = T + f (3) ρ du dt = q + tr (T D)+ρr (1) left-hand side ρ (1) v (3) u (1) so we have 5 equations for 14 unknown fields the system is highly under-determined closure relations required right-hand side T (6) q (3) Energy Balance 51 / 81 Closure Relations balance equations are universally valid closure relations describe the specific behavior of a material closure relations are often called constitutive equations Energy Balance 52 / 81

27 Material Science Rheology is the study of the flow of complex liquids or the deformation of soft solids. We are not going through this in great detail. For further information, consult the handout or take Martin Truffers Ice Physics 614 class We need to define a constitutive equation for the heat flux q and for the internal energy u a relationship between stress and strain, T = f (D) Constitutive Equations 53 / 81 Generalization Assume secondary creep incompressibility of ice uniform pressure does not lead to deformation T = pi isotropic + T deviatoric with the pressure p = 1/3trT The only non-straightforward step from τ = η (T, p, γ ) γ (18) to is the generalization of γ T = η (T, p, ) D (19) Constitutive Equations 54 / 81

28 Generalization is a relation between two tensors (i.e. T and D) T = η (T, p, ) D (20) material objectivity tells us that such a relationship must be independent of the chosen vector basis a second rank tensor has 3 scalar invariants trt = 0 for incompressible media σ e = II T I T = trt (21) II T = 1/2 trt 2 tr T 2 (22) III T = det T (23) is the effective stress role of III T is not clear, but lab experiments indicate that the third invariant is not important Constitutive Equations 55 / 81 Generalization Numerous lab experiments and field measurements suggest 1 η (T, p,σ e ) = 2A (T, p) f (σ e) (24) where A(T, p) = A 0 e Q/(RT ) Arrhenius law f (σ e ) = σ n 1 e power law Constitutive Equations 56 / 81

29 Generalization What s new? We have replaced by γ = η (T, p, τ ) τ (25) D = η (T, p, σ eff ) T (26) Constitutive Equations 57 / 81 Calorimetric Equation of State Internal energy, u depends linearly on temperature, T, u = u 0 + c (T T 0 ) (27) u 0 is a reference value at a reference temperature T 0 c is the heat capacity (a measure how much heat is needed to raise the temperature by a certain amount) we obtain du dt = c dt dt (28) Constitutive Equations 58 / 81

30 Heat Flux Fourier s law of heat conduction T j-1 T j T j+1 heat flows from higher to lower temperature this is the second law of thermodynamics q = k T t modified after Christophe Dang Ngoc Chan, wiki commons image the thermal conductivity k is a measure for how fast the heat spreads Constitutive Equations 59 / 81 Recap Balance Equations Flow Law mass 0 = v (1) momentum temperature ρ dv dt = T + p + f (3) ρc dt dt = q + tr (T D)+ρr (1) Glen s flow law T = 2 η D = A(T ) 1/n σ 1 n e D (6) now we have 15 equations for 15 unknown fields the system is now well-determined Recap 60 / 81

31 Large-Scale Dynamics Ice sheets are glaciers too! we will focus on the ice sheet part and, for now, ignore ice shelves ice is assumed to be incompressible, v = 0 Glaciers 61 / 81 The Incompressible Navier-Stokes Equation Momentum Balance ρ dv dt acceleration = T + p + ρg 2ρΩ v deviatoric isotropic gravity Coriolis force This is the Navier-Stokes equation for incompressible flow (29) still rather complicated do we really need all these terms? Scale Analysis is a power- and useful concept to assess the relative importance of terms Scale analysis 62 / 81

32 Scale Analysis Some typical values for an ice sheet horizontal extend [L] = 1000 km vertical extend [H] = 1km horizontal velocity [U] = 100 ma 1 vertical velocity [W ] = 0.1 ma 1 pressure [P] = ρg[h] =10 MPa time-scale [T ] = [L]/[U] =10 4 a 1 The aspect ratio is defined as = [H] [L] = [W ] [L] = 10 3 for an ice sheet The scaling argument for valley glaciers is almost the same Scale analysis 63 / 81 Scale Analysis Froude number The Froude number Fr is the ratio of acceleration and pressure gradient. In the horizontal we have Fr = ρ[u]/[t] [P]/[L] = ρ[u]2 /[L] ρg[h]/[l] = [U]2 g[h] and in the vertical Fr = ρ[w ]/[t] [P]/[L] = ρ[w ]2 /[L] ρg[h]/[l] = [U]2 g[h] The acceleration term is negligible Scale analysis 64 / 81

33 Scale Analysis Rossby number The Rossby number Ro is the ratio of acceleration and Coriolis force Ro = ρ[u]/[t] 2ρΩ[U] = ρ[u]2 /[L] 2ρΩ[U] and thus the Coriolis to pressure gradient is 2ρΩ[U] [P]/[L] = Fr Ro The Coriolis term is also negligible = [U] 2Ω[L] Scale analysis 65 / 81 Stokes Equation By neglecting both the acceleration term Coriolis term the Navier-Stokes equation simplifies to η v + v T p = ρg (30) gravitational force exerted on the ice is balanced by stress within the ice this is called the Stokes equation Stokes Equation 66 / 81

34 Temperature Equation Recall the equation for temperature ρ dt dt = q + tr (T D)+ρr (31) except for the uppermost few centimeters of ice, radiation is negligible ρ dt = q + tr (T D) (32) dt Energy Balance 67 / 81 Boundary Conditions Balance Equations mass v = 0 momentum η v + v T p = ρg temperature ρc dt dt (k T ) = tr (T D) Boundary Conditions To solve this system of equations, we need boundary conditions: mass (kinematic) momentum (dynamic) temperature (thermodynamic) Boundary Conditions 68 / 81

35 Ice-Thickness Equation H t = (q in q out )+a s a b = Q + a s a b (33) a s and a b are in the vertical direction the volume flux Q is defined as Q = Qx Q y h b = v x dz h b v y dz derivation on p this is the central evolution equation in ice sheet dynamics Boundary Conditions 69 / 81 Surface Dynamic Boundary Condition Continuity of the stress vector T n = T atm n (34) T atm is the Cauchy stress tensor in the atmosphere atmospheric pressure and wind stress are small compared to stress in the ice T n = 0 (35) this is a stress-free condition Boundary Conditions 70 / 81

36 Basal Dynamic Boundary Condition T n = T lith n (36) T lith is the Cauchy stress tensor in the lithosphere unfortunately, we have no information about the stress state of the lithosphere this boundary condition is not very useful empirical relationship is required relate bed-parallel shear stress τ b and bed-parallel basal velocity u b u b = f (τ b ) Boundary Conditions 71 / 81 Basal Dynamic Boundary Condition we can distinguish two cases ice is frozen to the bed if temperature below pressure melting point Tm ice is sliding if temperature at pressure melting point T m 0, T < Tm Dirichlet condition u b = f (τ b ), T = T m Robin condition despite 50 years of work (and lots of progress), this remains one of the grand unsolved problems in glaciology Boundary Conditions 72 / 81

37 Surface Thermodynamic Boundary Condition the surface temperature T s can be well approximated by the mean annual surface air temperature, as long as the latter is 0 C we therefore simply prescribe the surface temperature (Dirichlet condition): T = T s Boundary Conditions 73 / 81 Basal Thermodynamic Boundary Condition Cold Base the heat flux into the ice equals the geothermal heat flux q geo q n = q geo Temperate Base the ice is at the pressure melting point T m : T = T m Boundary Conditions 74 / 81

38 Thermomechanically-Coupled Glacier Flow Balance Equations mass v = 0 momentum η v + v T p = ρg temperature ρc dt dt (k T ) = tr (T D) Boundary Conditions mass momentum temperature h + v t x h + v x y h v y z = a s b + v t x b + v x y b v y z = a b T n = 0 u b = {0, f (τ b )} T = T s surface base surface base surface T = T m or q n = q geo base Boundary Conditions 75 / 81 Numerical Modeling the presented Stokes problem is solvable on a valley glacier scale but still very hard on a whole ice sheet scale for ice sheets, the shallowness (small aspect ratio) can be further exploited, this leads to a hierarchy of approximations Ed Bueler s Karthaus lecture is a good starting point Boundary Conditions 76 / 81

39 Basics and Notation We assume a fixed orthonormal basis e x, e y, e z Einstein summation convention says that double indices (here i) imply summation Nabla = e x x + e y y + e z z = e i x i Gradient the gradient of the scalar quantity g is thus g g = e x x + e g y y + e g z z = e ig,i = g/ x g/ y g/ z Math Stuff 77 / 81 Basics and Notation Divergence the divergence of the quantity g is g = g x x + g y y + g z z = g i,i Total Derivative or material derivative where v is the velocity dg dt = g t + v g Math Stuff 78 / 81

40 Basics and Notation Trace the trace of the n n matrixa is the sum of the diagonal elements tra = A 11 + A A nn = A ii Math Stuff 79 / 81 Recall from Vector Calculus Divergence Theorem φ n da = φ dv (37) ω ω The sum of all sources minus the sum of all sinks gives the net flow out of a region Reynolds Transport Theorem d dt ω φ dv = ω φ t dv + ω φv n da (38) What was already there plus what goes in minus what goes out is what is there Math Stuff 80 / 81

41 The Cauchy Stress Tensor T = t xx t xy t xz t yx t yy t yz t zx t zx t zz For example t ez = t xz t yz t zz is the stress vector of a cut along the xy-plane Math Stuff 81 / 81