Understanding Glacier-Climate Interaction with Simple Models

Understanding Glacier-Climate Interaction with Simple Models 65-4802-00L / FS 203 Rhonegletscher, Switzerland, 2007-202 (VAW/ETHZ) A proglacial lake is forming in the terminus area of Rhonegletscher (Furkapass, Wallis). Well visible is the white tarp covering the touristic ice cave at the far side of the glacier. Martin Lüthi Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie, VAW, ETH Zürich

Chapter Review of Glacier Dynamics This section is a short review on some essential aspects of glacier dynamics, which is mostly taken from the Physics of Glaciers I script. Good books on the topic include Cuey and Paterson (200) and Greve and Blatter (2009).. Field Equations To calculate velocities and stresses in a glacier we have to solve eld equations. For a mechanical problem (e.g. glacier ow) we need the continuity of mass and the force balance equations. Conservation of Mass The mass continuity equation for a compressible material of density is (in different notations) @ @t + @(u) @x + @(v) @y + @(w) @z @ @t + r (v) = 0 = 0 (.a) (.b) If the density is homogeneous ( @ @x i = 0) and constant (incompressible material @ = @t 0) we get, in dierent, equivalent notations Balance of Forces tr _" = _" ii = 0 r v = v i;i = 0 _" xx + _" yy + _" zz = 0 @u @x + @v @y + @w @z = 0 (.2a) (.2b) (.2c) (.2d) The force balance equation describes that all forces acting on a volume of ice, including the body force b = g (where g is gravity), need to be balanced by forces 3

Chapter Review of Glacier Dynamics acting on the sides of the volume. In compact tensor notation they read r + b = 0 ; (.3a) The same equations rewritten in index notation (summation convention) and in full, unabridged notation ij;j + b i = @ ij @x j + b i = 0 (.3b) @ xx @x + @ xy @y @ yx @x + @ yy @y @ zx @x + @ zy @y + @ xz @z + g x = 0 + @ yz @z + g y = 0 (.3c) + @ zz @z + g z = 0 These three equations describe how the body forces and boundary stresses are balanced by the stress gradients throughout the body. Rheology of Polycrystalline Ice Glacier ice usually considered as a nonlinear viscous uid, or more specically a power law uid. The most widely used ow relation for glacier ice is (Glen, 952; Nye, 957) _" ij = A n (d) ij : (.4) with n 3. The rate factor A = A(T ) depends on temperature and other parameters like water content, impurity content and crystal size. The quantity is the second invariant of the deviatoric stress tensor and dened by 2 := 2 (d) ij (d) ij = 2 (d)2 xx + (d)2 yy + (d)2 zz.2 Geometry Evolution + 2 (d)2 xy + 2 (d)2 xz + 2 (d)2 yz : (.5) If the surface of a glacier is described by a function F = 0, the following conditions holds @F @t + v rf = _ b (.6) The net balance rate _ b describes how much volume is added or removed from the surface, and has units m/a. 4

Understanding Glacier-Climate Interaction with Simple Models If the surface is written as a function z s of the horizontal coordinates F (x; y; z; t) := z s (x; y; t) z = 0, this reads @z s @t + v @z s x @x + v @z s y @y v z = _ b : (.7).3 Shallow Ice Approximation The vertical velocity distribution in a parallel sided slab n+ is u(z) = v x (z) = 2A n + (g sin )n H n+ (H z) {z } deformation velocity + {z} u b sliding velocity (.8) This is known as shallow ice equation, since it can be shown by rigorous scaling arguments that the longitudinal stress gradients @ xi @x i and @ yi @x i are negligible compared to the shear stress for shallow ice geometries such as the inland parts of ice sheets (except for the domes). 5

Chapter 2 Macroscopic Glacier Models Deep insights into the working of glaciers can obtained by omitting details of glacier dynamics, and concentrating on the main aspects of glacier response to climate. To this aim we will consider dierent Macroscopic Glacier Models. The idea is to simplify almost all aspects of glacier geometry and dynamics to elucidate the main processes. 2. Assumptions The volume V of a glacier depends on length L, width W and thickness H, and a shape factor f describing the fraction of the box W HL occupied by ice. It can be written V = f W HL : (2.) To treat glaciers as macroscopic entities means that one assumes the following Assumption : Details about ice thickness distribution are ignored. Assumption 2: Glaciers of dierent sizes are similar in shape. Þ Þ ¼ Þ Ä Î À Ä Î ¼ Ä Ü Figure 2.: Schematic geometry of a macroscopic glacier, and some important quantities. Assumption describes that one is only interested in total ice volume. Details of its distribution are lumped into the variables H (thickness) and f (shape). Assumption 7

Chapter 2 Macroscopic Glacier Models 2 describes an observation made in nature and from numerical models, that shapes of glaciers in steady state are often quite similar. Most glaciers have a characteristic longitudinal prole at the terminus. This similarity is (partially) captured in the shape factor f. The quantity H is a characteristic ice thickness scale which characterizes the whole glacier. 2000 2000 Elevation (m) 500 000 00 200 300 Elevation (m) 500 000 500 400 500 0 5 0 5 Distance (km) 600 500 0.006 0.024 0.048 0 5 0 5 Distance (km) Figure 2.2: Plots illustrating the similarity in shape for dierent glaciers on a simple bedrock. Left: Surface geometries for dierent equilibrium line altitudes between Z = 00 : : : 600 m (indicated below the glacier terminus). Right: Surface geometries for Z = 400 m and dierent mass balance gradients (indicated below the glacier terminus). The dashed line shows the glacier geometry for Z = 500 m and _g = 0:006 a for comparison. Symbols on horizontal lines: Solid dots indicate the horizontal location of the equilibrium line, upward pointing triangles the maximum ice thickness, and downward pointing triangles maximum ow velocities (from Lüthi, 2009). Further simplications of glacier geometry are Assumption 3: Glacier width W is constant. Assumption 4: Bed slope s := tan is constant. Assumption 5: The volumetric balance rate _ b(t) is given by a function that depends only on position x or on elevation z. (The volumetric balance rate has units units m 3 =( m 2 a) = m a.) balance rate B _ is simply the integral of b B(t) = Z L 0 _b(x; t) dx or _B(t) = Z L The volume change rate is given by the total mass balance rate 0 Glacier net _b(z s (x); t) dx : (2.2) dv dt = _ B: (2.3) 8

Understanding Glacier-Climate Interaction with Simple Models Assumption 6: Glacier length L(t) changes according to an evolution equation. There are two very dierent cases: either length L is proportional to V (i.e. immediate reaction), or an independent evolution equation is prescribed dl dt = ( HW dv dt a (L L a ) L proportional to V relaxation equation (2.4) The rst equation describes immediate reaction with a proportionality between L and V. The second equation is a relaxation relation with a as the relaxation time scale (the limiting case a! 0 is equivalent to immediate reaction). The climateadjusted length L a can be calculated by either using a volume-area scaling relation, such that L a = L a (V ) (e.g. Lüthi, 2009), or by linking it directly to climate through a climate sensitivity factor c L a = cz (e.g. Klok and Oerlemans, 2003). To proceed, one has to make an assumption about macroscopic rheology. This essentially amounts to prescribing how ice thickness H depends on the state of the glacier. Assumption 7: Characteristic ice thickness H is due to a macroscopic rheology. Three approaches have been used that are discussed in more detail below H =8 > < >: H 0 constant thickness; 0 =(gs) perfect plasticity; a L scaling relation (self-similarity): (2.5a) The constant thickness case is the simplest of all, and has been employed by Harrison (203) to investigate large changes. Perfect plasticity and scaling have been investigated by Messié (20). The models by Harrison (203) (constant thickness) and Klok and Oerlemans (2003) implicitly belong to the last case (with = and = 2, respectively). 2.2 Mass balance / Volume change Glaciers change their volumes through mass balance processes at the surface (mainly, there are more processes within and under the ice). Alpine type glaciers have a mostly elevation-dependent mass balance, due to the atmospheric temperature lapse rate. In what follows we assume a linear increase of net mass balance rate with elevation. Net balance at the equilibrium line altitude (ELA) is zero (by denition). Assumption 5b: Net balance rate is given by _ b = _g(z s (x) z ELA ) 9

Chapter 2 Macroscopic Glacier Models The mass balance gradient is dened as _g := d _ b=dz. For a constant mass balance gradient _g we can write total balance rate as _B(t) = W = W = _g W Z L Z0 L Z 0 L 0 _b(z s (x); t) dx _g (z s (x) z ELA ) dx using z s (x) = z b (x) + h(x) (z b (x) + h(x) z ELA ) dx (2.6) To simplify this expression we dene Z := z 0 z ELA and use the denition V = W Z L 0 h(x) dx = flhw to arrive at _B = _g [V + (z b z ELA ) LW ] : (2.7) In a steady state the total mass balance rate is _ B = 0 and thus (with bars denoting averages over the glacier length) z s = z ELA and V = (z ELA z b ) LW = (z s z b ) LW ; (2.8) and therefore H = z s z b (obviously). To further simplify things, we assume a constant bedrock slope s Assumption 4: Bed slope s := tan is constant. The bed elevation therefore is z b (x) = z 0 sx and mean bedrock elevation for constant slope is z b = z 0 sl=2, which leads to V = Wz ELA z 0 + s 2 L L = W s 2 L2 ZL ; (2.9) where Z := z 0 z ELA has been introduced for convenience (actually, this amounts to redening the vertical coordinate, cf. Fig. 2.). This Equation shows that for the assumed simplications, glacier volume depends non-linearly upon glacier length, and that mean ice thickness is given by (Oerlemans, 200) H = fh = Now we can also write total balance in the simplied form _B = _g W(Z + fh)l V W L = s 2 L Z : (2.0) 0 L2 s : (2.) 2

Understanding Glacier-Climate Interaction with Simple Models 2.3 V-model (block model) First, we discuss the V-model (Harrison, 203). This model has the characteristics of immediate reaction of length to changes in volume (assumption 6a), and constant ice thickness H 0 (assumption 7a, thus f = ). The geometry is just a block of ice, which gives rise to the name block model. Since length and volume are related through V = W LH 0 ; (2.2) the volume evolution can be written by replacing L in Equation (2.) dv dt = B _ = _g (Z + H 0 ) V s V : (2.3) H 0 2W H 0 2! This equation can be simplied by scaling the variables. We use the denitions P := + Z H 0 V b := 2W H 2 0 s t b := _g equilibrium line parameter volume scale, i.e. V? = V=V b time scale, i.e. t? = t=t b Denoting scaled variables with asterisk (?), Equation (2.3) reads dv? dt? = P V? V?2 = V? (P V? ) : (2.4) Equation (2.4) is the logistic equation describing resource-limited growth (also called the Verhulst equation). For a constant climate (i.e. P = const) and an initial volume V it has the analytic solution?0 V? (t? ) = + P V?0 This function is also known as the logistic function. P e P t? : (2.5)

Chapter 2 Macroscopic Glacier Models 2.4 Volume-Length Scaling The V-model has a major shortcoming: the thickness does not vary with changing glacier length. A straightforward way to remedy this is to use a more physical approximation for ice mechanics. The V-model is reminiscent of perfect plasticity, i.e. the ice either deforms without limit if the shear stress at the base b exceeds a certain threshold 0, or it does not deform at all. Glacier ice behaves dierently, it is well described as a viscous uid, with a stress-dependent viscosity (or alternatively: a strain-rate dependent viscosity). It belongs to the class of power-law uids, often encountered in geophysics; for details refer to the Physics of Glaciers I script, or Cuey and Paterson (200). The only formula we make use of is the ice ux through a vertical section of an inclined, parallel-sided slab. With a thickness H and surface inclination s s = dzs dx = s + dh dx, the shallow ice approximation (i.e. neglecting longitudinal stress gradients; Hutter (983)) yields gss Q(H) = 2 _" 0 (n + 2) 0 n H n+2 = _" 0 s n s H n+2 ; (2.6) where g and denote gravitational acceleration and ice density, and = 2:92 0 4 m 2. Equation (2.6) is the integral of Equation (.8). Using standard values for temperate ice n = 3, A = 25 MPa 3 a (Paterson, 999) and a reference stress 0 = MPa = 0 5 Pa leads to a reference strain rate _" 0 = 25 a. Total mass balance rate in the accumulation area depends on the volume, which we denote by V A = f A W GH, where the shape factor f A describes by how much the accumulation area deviates from a block shape. Similar to Equation (2.) we get _B A = _gw(z + f A H)) G + G2 s 2 = _gw ( f A )ZG + f A 2 sg 2 (2.7a) ; (2.7b) where use was made of the denition of the ice thickness H at the equilibrium line Z = sg + H =) H = sg Z : (2.8) In a steady state the ice ux through a vertical section at the equilibrium line equals the total mass balance of the accumulation area. Therefore, combining Equations (2.6) and (2.7b) evaluated at x = G and for n = 3, yields (Lüthi, 2009) H 5 = _g _" 0 s 3 s G ( f A )Z + f A 2 sg : (2.9) Position G and ice thickness at the equilibrium line H can now be determined numerically from the system of equations (2.8) and (2.9) for a given value of Z. 2

Understanding Glacier-Climate Interaction with Simple Models For the special case s > 0, Z = 0 (ELA at highest elevation of the bed), this system of equations has the unique solution s H = sg = _g f A s _" 0 s 2 3 : (2.20) For a general analytic approximation, we assume s > 0 and Z > 0 which corresponds to a mountain glacier geometry. To derive a volumelength scaling relation one has to make assumptions about the surface slope and the mean ice thickness of the accumulation area. Results from Full-Stokes-model runs (FS-model) show that the mean ice thickness in the accumulation area is similar to the ice thickness at the equilibrium line, i.e. f A 0:95. Under the additional assumption that the surface slope at the equilibrium line is equal to the bed slope (s s s), Equation ' (2.9) yields _g 5 2 H G 2 _" 0 s 2 5 : (2.2) Since the accumulation area ratio (AAR; the fraction of accumulation area to the total area) is almost constant at r 0:53 for all steady FS-model glaciers, G can be replaced by rl, leading to ' _g H 2 _" 0 f _g V ' 2 _" 0 5 r s2 5 L 2 5 ; (2.22) 5 r s2 5 L 7 5 = al : (2.23) The rst expression in parentheses depends on the parameters for mass balance and ice deformation, the second on geometry. The scaling exponent = :4 is similar to.36 obtained by Bahr et al. (997), although for a dierent geometric setting. Moreover the dependence of the scaling factor a on mass balance gradient _g, ice ow rate factor _" 0, and bedrock slope s is explicitly given in Equation (2.23). 3

Chapter 2 Macroscopic Glacier Models 2.5 LV-model The LV-model is a simplied representation of glacier dynamics as a two-variable dynamical system in the variables length L and volume V (Lüthi, 2009). The dynamical system is formulated for unit width, and reproduces the essential inuence of mass balance and ice dynamics on glacier geometry on a macroscopic scale. Figure 2. illustrates the building blocks of the dynamical system: two reservoirs of volumes V A and V B which are linked by a ux element located at horizontal coordinate G. The resulting dynamical system is (Lüthi, 2009, Eq. 40) dv hv dt = _g " + ZL dl dt = V a a L2i s # ; (2.24a) 2 L : (2.24b) Equation (2.24a) is the same as Equation (2.3) in the V-model. Equation (2.24b) is a relaxation equation for the current glacier length L with time constant a (in years; a should not be confounded with the volume time scale v ). The steady state length for the current volume V is determined by the volume-length scaling relation given in Equation (2.23). This dynamical system (2.24) is driven by a forcing in the term Z(t) = z 0 z ELA (t). 4

Understanding Glacier-Climate Interaction with Simple Models List of Symbols _b volumetric mass balance rate m a _g vertical gradient of mass balance rate _g = @ b _ a @z _B total mass balance rate m 3 a s bed slope s = tan H glacier thickness m L glacier length m V glacier volume m 3 W glacier width m Z elevation span of accumulation area Z = z 0 z ELA m Z? elevation span of ablation area Z? = z ELA z L m x horizontal along-ow coordinate axis m y horizontal across-ow coordinate axis m z vertical coordinate axis m 5

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