1 Glaciers and climate

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1 1 Glaciers and climate 1.1 Equilibrium length For a glacier to be in equilibrium with climate the (integrated) mass balance must be equal to zero. For a glacier of a uniform width and length l this implies = l where a is the (local) mass balance (usually measured in meters of water equivalent per year). Here we consider the case when a is a function of distance only, i.e. a at a given location is assumed not to change with surface elevation (how realistic is this assumption for alpine glaciers? And what is the situation for ice caps?). If the mass balance a changes at t = t by the constant value a across the whole glacier, then the new equilibrium length of the glacier will become l + l where We can write this as = l = l+ l a dx, (a + a) dx. l l l a dx + a dx + a dx + a dx. l l The first term above is zero. The last term is of second order, and provided /l 1 and a/a(l) 1, much smaller than the second and the third term. We can therefore write or = a(l) l + l a, l = l a a(l). (1) This gives us a relationship between the change in glacier length ( l) and the original glacier length l, the local mass balance at the snout (a(l)), and the (spatially constant) shift ( a) in surface mass balance. 1.2 Volume time scale We can get a rough lower estimate for how long it will take for a glacier to reach a new steady state following a perturbation a in surface mass balance. If the change in volume between these two steady states is V, then it will take the time t V = V A a Table 1: Symbols a (specific/local) mass balance (m/a) s surface elevation (m) b bedrock elevation (m) h = s b thickness (m) t V volume time scale (a) t R reaction time scale (a) t r relaxation time scale (a) t p phase, or wave propagation, time scale (a) w glacier width (m) 1

2 Thickness Change Elevation h( x,t) h (x) l h(x,t) l(t) h( l,t) x l l + l(t) Figure 1: Changes in ice thickness following a perturbation in surface mass balance. where A is the glacier area to fill this volume. This is the definition of the volume time scale. Observations show that most of the volume change happens in the vicinity of the snout with, in comparison, little changes in ice thickness in the accumulation area. A rough estimate for the volume change V, assuming a constant glacier width w, is therefore V h w l, (2) where h is some ice thickness scale. This could for example be the ice thickness at the equilibrium line, h = h eq, or alternatively we could use h = V/ A, where A is the glacier area. The accuracy of the volume estimate (2) will vary greatly. (Can you think of a situation where this is a very inaccurate estimate.) We now can write t V = hw l wl a and using Eq. (1) we arrive at t V = h/a(l). This gives us a lower estimate for the reaction time of glaciers. Inserting values typical for Alpine glaciers suggest that the reaction time is at least on the order of a few decades. The volume time scale is a lower estimate of the reaction time because we assume that the mass needed to fill the volume V enters the volume without a delay. In reality, the mass-balance perturbation will be felt across the whole glacier and some of the mass added in the accumulation area will need to be transported to the ablation area where the biggest change in volume takes place. As we will see later, there are two timescales that are primarily responsible for the redistribution of mass across a glacier. These are the relaxation time scale t r and the phase velocity time 2

3 Figure 2: Surface mass balance distribution. scale t p. If t V is considerably larger than either of these time scales, the the reaction time scale t R is approximately given by t V. The current understanding is that this is often the situation, in which case t V is a good estimate for the response time of glaciers. 1.3 Glacier response to a perturbation in surface mass balance To illustrate the response of glaciers to climate change we model numerically (using SIA) the response of an alpine glacier to a step change in surface mass balance. Initially the surface mass balance (a s ) is prescribed as a function of distance as a s = a + a 1 x where a = 5 and a 1 = 2a/ The glacier is allowed to reach a steady state, and the mass-balance distribution is then shifted everywhere by the constant amount 1 m/yr. For t 1 yr the mass balance is given as a s = 1 + a + a 1 x. These two mass balance distributions, i.e. the initial one for t < 1 and the perturbed one for t 1 are shown in Fig. 2 in blue and red, respectively. The initial equilibrium length, i.e. for t < 1, of the glacier l is given by l (a + a l x) dx, and we find that l = 2a /a 1 = ±5 km. The final equilibrium length (l 1 ) is l1 (a + a l x + 1) dx, or l 1 = ±6 km. We allow the glacier to grow for 1 years from an initially ice-free state until a steady state is reached. A steady-state is reached after approximately 5 years, and the resulting steady-state geometry is shown in Fig. 3. Once the mass-balance perturbation is applied for t 1 yr the glacier grows towards a new steady state. The resulting changes with time in length and volume are shown in Fig. 5. Note that the volume starts to change immediately once the mass-balance perturbation is applied. The greatest rate-of-change in volume (dv/dt) is right at the beginning and the tailors gradually off towards zeros as a new steady-state is reached. 3

4 Figure 3: Glacier geometry (upper panel) and surface velocity and surface mass balance (lower panel) for the year 999. The length reacts with a slight delay and the greatest rate-of-change in length (dl/dt) is only reached a few years after the mass-balance perturbation is applied. 1 The (integrated surface) mass balance M = l l a s dx, is shown in Fig. 5. The mass balance is zero before the perturbation is applied, a necessary condition for a steady state, and again zero once a new steady state is reached. (It appears that the mass balance (M) is not a particularly good indicator of climate change.) There is an initial step change in M. In response to the changes in glacier geometry, the mass balance then decreases with time and slowly approaches zero as a new steady-state is reached. The volume time t V can be estimated from Fig. 3 as t V = 4/( 5) = 8 years. As we can see from Fig. 5 the e-folding time is about 1 years, so the volume time scale, t V, seems to give a reasonable good estimate of the reaction time. We could also estimate a reaction time by dividing the length of the glacier with a typical surface velocity. This approach gives 5 km/1 m/yr or about 5 years. This time scale is considerably longer than the actual reaction time (Note: 1 The evolutionary equation for the thickness (assuming flat base with b = and s = h) is t s = 1 ( ) n + 2 x s n+2 x s n 1 x s + a This is a degenerate nonlinear diffusion equation with a singularity at s = and discontinuity x s/ x s at the ice divide. One can show (see Fowler s book Mathematical Geoscience) that the the surface slope at the margin is finite for a retreating glacier, but infinite for an advancing glacier. for t x m <, where x m is the position of the ice margin, and for t x m >. x s (a/ t x m ) x s (x m x) (n+1)/(2n+1) 4

5 Figure 4: Glacier length and volume as a function of time following a perturbation in (specific) surface mass balance at t=1 (year). Figure 5: Integrated surface mass balance as a function of time following a perturbation in (specific) surface mass balance at t=1 (year). 5

6 the calculations were done for a linear medium and the kinematic wave velocity for long waves is then only twice the surface velocity, giving an estimate of 25 yr.) The time scale obtained by dividing thickness with surface velocity is therefore considerably longer than the (modelled) reaction time and apparently not a good estimate of the time it takes for glaciers to reach a new steady state after a perturbation in surface mass balance. 6