Analysis of Alpine glacier length change records with a macroscopic glacier model

Analysis of Alpine glacier length change records with a macroscopic glacier model The length change record of 91 glaciers in the Swiss Alps was analyzed with a novel macroscopic glacier model (LV-model). Based on a history of equilibrium line variations, synthetic length change data were calculated. From the LV-models matching best the measured length changes, characteristic parameters were obtained. The volume time scale thus determined ranges from 5 to 170 years for glaciers of different slope and length. The analysis shows that the observed glacier length changes cannot be reproduced with an equilibrium line variation based on temperature and precipitation alone. The equilibrium line has to be lowered by 100 to 200 meters during several phases of the Little Ice Age (in the time span 1650 to 1850) to obtain observed glacier responses. Such an effect might be attributable to either higher winter precipitation in the Alps, or to radiation forcing.


Martin P. Lüthi and Andreas Bauder, Zurich 1 Introduction
The record of glacier length changes in the Swiss Alps, measured during the last century (Glaciological Reports 2009), is unique in its length, its spatial coverage and its variety of glacier geometries.The collective data set contains a wealth of information about past changes in glacier mass balance, and thus climate, albeit in a indirect manner.These data can be viewed as a set of sensors probing climate, where each sensor has a different response to the external forcing.
The position of a glacier terminus depends on two processes: advection of ice into the terminus area, and melting of ice at the surface.If both processes are of the same magnitude, the glacier terminus geometry remains unchanged.Upon a sudden change in mass balance, the terminus geometry reacts immediately due to increased or decreased melting, and with some delay until glacier dynamics changes the mass transport into the terminus area (e.g.Nye 1963).For an oscillating climate, this delay in response depends on the frequency of the mass balance changes, and can be out of phase for frequencies lower than the volume time scale (Hutter 1983;Lüthi 2009;Nye 1965a).
Several studies have used glacier length changes to infer climate history (e.g.Klok & Oerlemans 2003;Nye 1965b;Oerlemans 2001;Oerlemans 2005).Steiner et al. (2005) and Steiner et al. (2008) used reconstructed climate data to drive a neural network trained on a glacier length record to analyze length changes, to infer climate sensitivity, and to predict the future evolution of several glaciers.The studies by Harrison et al. (2003) and Oerlemans (2007) are quite similar in scope to the present study, although with different approaches to glacier dynamics, and for a considerably smaller number of glaciers.
In this contribution, the length response of 91 glaciers from the data set of glacier length changes from the Swiss Glacier Monitoring Network (Glaciological Reports 2009) are analyzed.To this aim, a macroscopic glacier model is used which is formulated as a dynamical system in the variables length and volume (Lüthi 2009).From the model results, parameters are obtained which are characteristic for these glaciers, the most important of which are the volume time scale as well as constraints on the Little Ice Age equilibrium line history.

Length change data
The data set of glacier length changes from the Swiss Glacier Monitoring Network is used in this study (Glaciological Reports 2009).This publicly available data set contains 120 glacier length change records with yearly measurements.For 27 glaciers there are time series of more than 100 years, and 3 glaciers have been measured for more than 120 years (Fig. 1).In this study, the length changes for 91 of these glaciers are analyzed, which have a homogeneous data set covering at least 35 years.

LV-model
A macroscopic representation of glacier response to climate is used, formulated as a two-variable dynamical system in the variables «length» L and «volume» V (Lüthi 2009).The dynamical system reproduces on a macroscopic scale the essential influence of mass balance and ice dynamics on glacier geometry, as represented with the variables L and V. Figure 2 illustrates the building blocks of the LV-model: two reservoirs of volumes VA and VB which are linked by a flux element located at horizontal coordinate G.The ice flux through a vertical section at the equilibrium line is determined by ice thickness and surface slope according to the shallow-ice approximation (Hutter 1983).Local mass balance rate is parametrized as a linear function of elevation.From these assumptions, a system of two ordinary differential equations (ODE) can be derived (Lüthi 2009, Eqs. 40) ∂b where g = ∂z is the vertical gradient of mass balance rate (in units of meter ice thickness per year), mb = tan b is bedrock slope, ta is the relaxation time constant for the length adjustment, and parameters a and m = 7/5 describe the volume-length scaling relation.The scaling parameter a depends explicitly on g and b (Lüthi 2009, Eq. 21).The dynamical system (Eq. 1) contains an external forcing term in Z(t) = z0 -zELA(t), where z0

Analysis of Alpine glacier length change records with a macroscopic glacier model
is the highest point of the bedrock (Fig. 2), and zELA(t) is the time dependent equilibrium line altitude (ELA).In vicinity of a steady state, the LV-model is equivalent to a linearly damped harmonic oscillator (Harrison et al. 2003) which is slightly over-damped.The dynamical system (Eq. 1) was solved numerically with the PyDS-Tool toolkit (Clewley et al. 2004).

Equilibrium line history
In the LV-model, local mass balance rate at the glacier surface is prescribed as a linear function of elevation b(z) = g(z -zELA), with a constant vertical gradient g of local mass balance rate.A changing climate is hence parametrized as a change in ELA, which appears as forcing in the Z term of the LV-model (Eq.1a).To calculate glacier length changes, a history of ELAs was prescribed, which is based on a reconstructed record of temperature and precipitation of Europe since 1600 (Casty et al. 2005).A spatial average of monthly data for 9 grid points centered in the Gotthard area was used.
To obtain ELA variations from temperature and precipitation, a bi-linear relation between temperature T, precipitation P and ELA was assumed of the form (2) Pref which is equivalent to a standard climate-ELA relation (Ohmura et al. 1992, Eq. 1) if the derivatives ∂T ∂P ∂z and ∂T are constant.The values of the constants were determined by fitting the parametrized ELA changes to reconstructed ELA variations for several Swiss glaciers (Huss et al. 2008;Huss 2009).The best agreement between the climate and ELA reconstructions was found for summer (JJA) temperature, and yearly average precipitation, with the constants a = 2738 m, b = 101 m K -1 , c = 200 m and Pref = 2000 mm.
As will be shown below, the ELA reconstruction is not suitable to produce any of the big and rapid Little Ice Age glacier advances observed between 1650 and 1850.To achieve a match of measured length changes before 1910, the ELA had to be lowered by 100 to 200 m for certain periods within the time span 1650 to 1850.

Model results
The response of a glacier to climate forcing depends on its geometry, which in the LV-model is simply parametrized as bedrock slope b and vertical extent Z of the accumulation area.Driven by a history of ELA changes, glacier length changes were calculated with the LV-model in the following manner: The model glaciers were initialized to a steady state in the year 1600 for each set of parameters g, b and Z.The model was driven with an ELA history calculated from temperature and precipitation from the climate reconstruction (Eq.2).The ELA history used for the time span after 1880 is shown in Figure 3c, and the complete history in Figure 4b.
The influence of the geometry parameters on glacier length response is investigated in Figure 3.The ELA history shown in panel 3c was used to drive the LVmodel (Eq. 1) for different values of b (Figure 3a) and Z (Figure 3b).It is immediately obvious that flat glaciers and glaciers with a small vertical extent show a very smooth response, and therefore have long response times.On the other hand, steep glaciers, and glaciers spanning a high elevation difference show a large and fast response to short-term fluctuations of the ELA.
The model results in Figure 3 look similar in character to the measured length change records from the data set (Figure 1).For a quantitative comparison, an optimization procedure was used to find a set of model parameters (g, Z, b) which produces the best-match-ing length change history.With a set of parameters, the dynamical system was integrated forward in time for each of the ELA histories shown in Figure 4b as driving function (many more unsuccessful attempts are not shown).Each of the measured length change records was then compared to the responses of the model glaciers to find the closest match between real and model glaciers.The best overall agreement was obtained with g = 0.008 a -1 , which was adopted for all glaciers.The best fitting model glacier then yields the characteristic parameters Z and b, and from these the derived quantities «model glacier length» L and «volume time scale» tv can be determined.
The modeled glacier response is strongly influenced by the climate history assumed between 1650 and 1850, a climatic episode termed the «Little Ice Age» (LIA).For most glaciers, especially those with a long response time, it is impossible to obtain a reasonable fit between modeled and measured length changes if the model is forced with ELA variations according to the climate reconstruction of precipitation and temperature alone.Since mass balance also depends on variation of solar radiation (e.g.Huss et al. 2009), and inspired by the reconstruction of the radiative forcing for the time span considered (e.g.Crowley 2000; Steinhilber et al. 2009), the ELA was lowered during certain phases of the LIA.Taking Grosser Aletschgletscher as an example, Figure 4 shows the modeled response for three different LIA climates which differ only by a constant offset of the ELA during certain time spans.A reasonable agreement for the length change records of most glaciers could be obtained for an ELA low-ered by 100 m between 1680 and 1720, and by 150 m between 1800 and 1850 (solid line in Figure 4b), which was adopted for the rest of this study.
Length change data for 91 glaciers of the Swiss Glacier Monitoring Network contain enough homogeneous data points to be fitted with modeled length changes.These best-fitting model glaciers capture the essential dynamics of a glacier terminus, and yield values for b, Z and the volume time scale tv. Figure 5 shows how well the individual glacier length records could be fitted with the LV-model.Table 1 lists characteristic quantities and model parameters for the glaciers.Also listed are inferred volume time scales which range from 5-20 years for very steep glaciers (e.g.Rosenlaui, Orny, Trient), 130-140 years for Grosser Aletschgletscher, and up to 160-180 years for several smaller glaciers.The increasingly large uncertainties of determined volume time scale for longer-timescale glaciers is due to the relatively short sampling interval of glacier response, as compared to the volume time scale.
The method works surprisingly well even for glaciers that would appear as problematic: heavily debris-covered glacier tongues (e.g.Unteraar, Zmutt; whereas Mont Durand cannot be fitted), glaciers that went through a strong topographic break during the sampling interval (e.g.Blüemlisalp, Eiger, Palü, Rhone, Tiatscha, Turtmann; whereas Mont Fort cannot be fitted), and glaciers that were affected by hydraulic dams (Unteraar, Oberaar, Gries) and natural lakes (Roseg, Gauli, Trift), where formation of a proglacial lake lead to temporarily fast retreat.It is noteworthy, that not only long glaciers (Grosser Aletschgletscher, Gorner, Otemma) have a long volume time scale, but also several smaller glaciers (e.g.Paradies, Roseg, Cheillon).The theory used to derive the LV-model explains that the volume time scale depends only on the «activity index» z like tv -1 = g(z -1) (Lüthi 2009).This parameter, defined by

Climate
The glacier length variations during the last 150 years of all 91 investigated glaciers can be explained with a single history of ELA variation, and a constant mass balance gradient of g = 0.008 a -1 .The similar variation of the ELA throughout the whole Swiss Alps, despite large differences in local climate, is attributable to the strong dependence of ELA on temperature.Air temperature anomalies are well correlated in the Alpine area (e.g.Casty et al. 2005).
The measured glacier length changes could only be reproduced with the LV-model if the ELA history (calculated from a reconstructed temperature and precipitation history) was considerably altered during certain periods of the LIA.The marked advance of most glaciers between 1830 and 1850 cannot be reproduced without such an ELA alteration.
The necessity to alter the ELA-history can have several reasons: • Reconstructed air temperature and precipitation rely mainly on data from low elevation stations, especially before the 20 th century, which might result in a misrepresentation of the reconstructed climate at high elevations.• The time spans of altered ELA correspond to phases Fig. 5: Modeled length changes under the same climate are shown for 91 glaciers with solid lines (vertically shifted for clarity).Measured length changes are indicated with dots.At the end of each line, the 3 letter abbreviation of the glacier name (cf.Table 1) and the volume time scale are given.