Physical energy balance and degree-day models of summer ablation

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Physical energy balance and degree-day models of summer ablation Powered By Docstoc

Physical energy balance and degree-day models of summer
       ablation on Langjökull ice cap, SW-Iceland

                        Sverrir Guðmundsson
                           Helgi Björnsson
                            Finnur Pálsson
                        Hannes H. Haraldsson

                Science Institute, University of Iceland
                 National Power Company of Iceland


1. INTRODUCTION........................................................................................................................ 2
2. OBSERVATIONS........................................................................................................................ 2
3. METHODS ................................................................................................................................... 4
   3.1. ENERGY BUDGET DURING THE MELTING SEASON ..................................................................... 4
   3.2. DEGREE-DAY MODELS ............................................................................................................. 5
4. RESULTS ..................................................................................................................................... 6
   4.1. VARIATION OF THE ENERGY BUDGET DURING THE ABLATION SEASON ..................................... 6
   4.2. VARIATION OF THE ENERGY BUDGET WITH ELEVATION ........................................................... 9
   4.3. EMPIRICAL ABLATION MODELS (DEGREE-DAY) ...................................................................... 11
   4.4. SENSITIVITY OF THE SUMMER MELTING TO CLIMATE CHANGES .............................................. 13
5. CONCLUDING REMARKS .................................................................................................... 18
ACKNOWLEDGEMENTS .......................................................................................................... 18
REFERENCES............................................................................................................................... 19

Langjökull is the second largest ice cap in Iceland (925 km2 in area), located at ~64.7°N and
~20.4°W in southwest Iceland (Fig. 1). The range in elevation is from 450 to 1450 m a.s.l., with an
average height of 900 m a.s.l. The surroundings of Langjökull comprise lava, sand, and proglacial
lakes. The major rivers draining the ice cap are Hvítá in Borgarfjörður and Hvítá in Árnessýsla. A
significant part of the glacial meltwater drains directly into groundwater (Sigurðsson, 1990).
        Measurements of annual summer and winter balance ( bs and bw ) have been conducted at
points along flowlines on Langjökull since 1996 (Björnsson and others, 1997; 1998a; Pálsson and
others, 2000, 2001). Each year snow cores have been drilled in April-May through the winter layer
and profiles of density measured. The summer balance was derived from observations in
September-October of profiles of density and readings at stakes and wires drilled into the glacier in
April-May (see Björnsson and others, 1998b). The glacial-meteorological observations were
initiated in 2001. Two automatic weather stations (AWSs) at the southern outlet Hagafellsjökull
(Fig. 1) collected meteorological data that allowed calculation of the surface energy budget. Two
AWSs, located outside the ice cap (Fig. 1), were used to obtain empirical relationships between
temperatures within and outside the glacier.
        This paper presents observations of
melting and calculations of the energy balance
at Hagafellsjökull during the ablation season
2001. The energy balance is related to the mass
balance of the glaciological year 2000-2001.
Simple empirical degree-day models are used to
relate the summer ablation to temperature
observations. Both the energy balance and the
degree-day models are used to evaluate glacial-
hydrological response to possible climate

Two automatic weather stations (AWSs) were
operated at Hagafellsjökull during the ablation
season 2001: at 490 m (G490) and 1060 m a.s.l.
(G1060) (Fig. 1; Guðmundsson and others,
2002). The AWSs measured the incoming ( Qi )
and outgoing ( Qo ) solar radiation, incoming
( I i ) and outgoing ( I o ) long-wave radiation, and
wind speed ( u ), wind direction ( WD ), air
temperature ( T ) and relative humidity ( r ) at 2
m above the surface, and the melting rate ( h ) of
the surface by a sonic echo sounder (Fig. 2;
                                                            Figure 1. Location of automatic weather stations.
Table 1). Air temperature was also recorded by              (a): Locations at Hagafellsjökull (G490 and
two AWSs outside the glacier: at 299 m a.s.l. on            G1060), at Söðulhólar (S299) and north of
Söðulhólar (S299) and 474 m a.s.l. north of Mt.             Skjaldbreiður (S474), see Table 2 for locations
Skjaldbreiður (S474) (Fig. 1).                              and elevations. Black dots: points of mass balance
          The meteorological instruments were               stakes. Langjökull (L) and Vatnajökull (V) ice caps
                                                            are shown on the inset map of Iceland. (b): Cross-
operated on the glacier from the late April to the          section of the profile in (a).
beginning of October 2001, fully covering the
ablation season (Table 2). All instruments were

                                                          calibrated in Reykjavík, in April 2001 and 2002.
                                                          The AWSs were visited at the end of June and
                                                          beginning of September 2001, to ensure that
                                                          they were working properly. Data were missing
                                                          at both the AWSs for about 10 days at the end of
                                                          May (Table 2) due to an overflow in memory
                                                          storage. Further, continuous records of the
                                                          melting rate were not available after June 26 in
                                                          the upper station and after August 11 at the
                                                          lower station due to malfunction in the sonic
                                                          echo sounders. However, observations of the
                                                          total accumulated ablation up to September 2 at
Figure 2. Upset of the AWSs at Hagafellsjökull.           the upper station, and October 5 at the lower
The meteorological instruments were mounted on            one, were used to linearly reconstruct the melt
a mast that followed the melting surface (always at
~2 m above the surface) and the sonic echo                rate patterns over the periods when the sonic
sounder on a mast, drilled several metres into the        records were missing (Fig. 3a).
glacier (independent to changes of the glacier                      Air-pressure and vapour-pressure were
surface). Parameters are defined in section 2 and         not measured on the glacier. The air-pressure
the accuracy of instruments in Table 1.                   ( P (a ) ) was calculated as a sample mean of an
                                                          observed air-pressure ( P(a0 ) ) at several
                                                          reference stations outside the glacier by the
                                                          experimental equation

                   0.0065(a − a 0 )
P(a) = P(a 0 ) 1 −                                                                        (1)
                    T (a 0 ) + 273

where a and a0 are the altitudes of the AWS on the glacier and the reference station, respectively.
This relationship has been proven to apply successfully at various locations outside and inside the
Vatnajökull ice cap (e.g. Björnsson and Guðmundsson, 1997; see Fig. 1 for location). The vapour-
pressure was calculated as e = r ⋅ e s / 100 , where r is the measured relative humidity and the
saturation vapour-pressure in Pa is given by the equation

es = 611.213 exp 17.5043                .                                                 (2)
                              T + 241.2

Table 1. Observed meteorological parameters at the automatic weather stations; instruments and accuracy.
Parameters are defined in section 2.
                     Observation               Equipment                   Accuracy
                        T, r                Vaisala HMP35                  0.2°C, 2%
                       WD , u                  R.M.Young                  5°, 0.1 m s-1
                   Q , Q , I , I          Kipp & Zonen CNR1            3%, 3%, 3%, 3%
                      i   o   i   o
                          h                           SR50                 max(1 cm, 0.4%)

Table 2. Coordinates of the AWS´s and observation periods. See Fig. 1 for location.
                          Location in °     Elevation        Observation             No data
                       latitude, longitude   m a.s.l.       interval 2001           available
          G1060      64.592 N, 20.425 W      1060           19.4. – 5.10.        21.5. – 28.5.
           G490      64.494 N, 20.437 W       490           20.4. – 5.10.        21.5. – 29.5.
           S474      64.448 N, 20.676 W       474           1.1. – 31.12.
           S299      64.341 N, 20.909 W       299           1.1. – 16.12.

The radiation components were measured
directly and the turbulent energy exchange
calculated from one-level measurements of
wind, temperature and humidity. Empirical
models were derived relating the surface-
melting rate to air temperature (degree-day

3.1. Energy budget during the melting
The energy budget at the melting surface can
be written as

Mc = R + Hd + Hl                                                  (3)

where R = Qi − Qo + I i − I o is the net-
radiation calculated from the observed
radiation components, and H d and H l are the
vertical eddy flux of sensible and latent heat,
respectively. We assume that the heat supplied
by rain is negligible. An eddy flux model
                                                                            Figure 3. Energy budget at Station G490 along with
taking account of the stability of the boundary                             surface melt rates. (a): Surface melting. (b): Daily
layer (Monin-Obukhov) was used to calculate                                 values of the total energy supplied for melting,
 H d and H l (e.g. Björnsson, 1972;                                         calculated through Eqs. (3) and (7). (c): The time
                                                                            series in (b), filtered by a three-day moving average.
Guðmundsson, 1999, p. 3-6). The model can                                   (d): Energy components filtered by a three-day
be simplified for one-level measurements as                                 moving average. The parameters are explained in
                                                                            section 3

                                 T ( z) − T ( z0 )
H d = ρ1c p k 02 u ( z )                                    ,                                                                 (4)
                           (ln( z ) − ln( z 0 ) + β ) 2

                               ρ1             e( z ) − e( z 0 )
H l = Lv k 0 u ( z )(0.622
                                    )                                   ,                                                     (5)
                                P                               z
                                        (ln( z ) − ln( z 0 ) + β ) 2

where T (z ) , u (z ) and e(z ) are the air temperature in °C, wind-speed in m s-1 and vapour-pressure
in Pa, respectively, at the height z above the surface. The roughness parameter of the surface ( z 0 )
is defined as the height where the wind-speed is zero. For a melting glacier surface T ( z 0 ) ≈ 0 °C
and e( z 0 ) ≈ 611.213 Pa. The parameter k 0 = 0.4 is the von Kármán constant, c p = 1010 J kg-1K-1
is a specific heat capacity of air at constant pressure, Lv = 2.5 ⋅ 10 6 J kg-1 is the latent specific
evaporation heat, and the constant β lies within the interval 6 to 7. The air density is given as
 ρ1 = ρ 0 ( P / Po ) , with ρ 0 = 1.29 kg m-3, P0 = 1.013 ⋅ 10 5 Pa and P as the air pressure in Pa. The
Monin-Obukhov length is expressed for one-level measurements and z >> z 0 as

L = −A +                                                                                                            (6)

with A = βz /(ln( z ) − ln( z 0 )) and B = ( g / T0 )(T ( z ) / u 2 ( z ))(ln( z ) − ln( z 0 )) , where g = 9.8 m s-2 is
the acceleration of gravity and T0 = 273.15 K.
        Values of the roughness coefficient z 0 Table 3. Applied values of surface roughness ( z 0 ).
for various surface conditions (Table 3) were                                 z 0 mm     ln( z 0 )
adopted from experience of numerous detailed
                                                   New snow                      0.1      -9.2
studies of the energy budget on Vatnajökull        Melting snow/firn             0.7      -7.3
during the period 1996-2000 (unpublished data;     Ice in an ablation zone        1       -6.9
see Fig. 1 for location) using one- and two-level
Monin-Obukhov eddy flux models with stability
        The total energy supplied for melting, was estimated directly from the observed daily
melting rate ( h ) as

M m = h ⋅ Ll ⋅ ρ                                                                                                    (7)

where ρ is the glacier surface mass density and Ll = 3,3 ⋅ 10 5 J kg-1 the latent heat of melting.
Thus, we estimate the total energy with two independent and complementary methods with aid of
Eqs. (3) and (7), i.e. from the observed weather parameters and direct observation of the melting
rate. The high consistency between the calculated energy with the two methods, both variations
and the total amount (Fig. 3b-c, Table 4) supports the evaluation of the roughness coefficients in
Table 3.

3.2. Degree-day models
We have considered four regression models between observed specific ablation rate ( a s ) and the
number of degree-days. The models are written as

Table 4. Monthly averages of the energy components and weather parameters. Data were missing at both
G490 and G1060 for ~10 days at the end of May (Table 2). See definition of parameters in sections 2 and 3.
Month            H                H                      R             M        Q        α      T       u
                      l                  d                                         c        i
         W m-2        % of       W m-2       % of       W m-2        % of      W m-2     W m-2     %       °C      m s-1
                      Mc                     Mc                      Mc
May        0.1             0      17.2        30        40.3          70        57.4      183      67      2.3     5.3
June      13.8             7      50.0        24        145.9         69       209.8      217      20      3.8     5.3
July      24.3            10      60.3        26        147.8         64       232.5      164       7      5.4     4.5
Aug.      30.8            13      76.5        32        128.2         55       235.5      151      7       5.6     5.0
Sept.     26.8            17      67.2        42        64.6          41       158.6      78       7       4.9     4.7
May       0.3              2      1.0          6         14.4         92       15.7       227      83     -1.3     7.0
June      -0.7            -1      7.1         13        46.3          88       52.7       280      72     0.7      5.5
July      6.7              8      18.0        20         64.7         72       89.3       215      64     2.1      4.2
Aug.      10.6             9      24.0        21         80.3         70       115.0      174      37     2.4      5.0
Sept.      8.1            15      17.3        31         30.2         54        55.6       77      38     1.1      6.2

a s = ddf 1        TG+ ,                                                                                (8)

a s = ddf 2        (TS + d ) + ,                                                                        (9)

a s = ddf 3         TS +    (hG − hS )          ,                                                      (10)
              t1         ∆h

a s = ddf 4        (TS + γ (hG − hS ))+ .                                                              (11)

The sums are taken over the period from day t1 to t 2 of the ablation season where ddf 1 to ddf 4
are scaling coefficients that remain constant with time but vary for snow and ice/firn, TG is
temperature at an elevation hG on the glacier, TS is temperature at a weather station outside the
glacier of elevation hS (here S474) and                 stands for degree days. Mean temperature difference
between these sites is d = TG − TS . Thus, (TS + d ) + is an estimate of TG . Equation (11) ignores
the high lateral temperature gradients between the melting glacier surface and the surrounding low
albedo areas and uses the constant lapse rate γ = 0.6 × 10 −2 °C m-1. The term ∆T ∆h is the
temperature gradient between two stations outside the glacier (here S299 and S474), and is often
used to estimate γ .

We relate the summer balance to the weather parameters and surface albedo. The total energy
supplied for melting was derived at the two stations on Hagafellsjökull from the observed ablation
and calculated from the meteorological observations. Calculations were done at daily, monthly and
seasonal time scales. The obtained energy components are used to evaluate the performance of the
empirical degree-day models for calculation of the ablation. The sensitivity of the energy budged
to possible future temperature changes are investigated.

4.1. Variation of the energy budget during the ablation season
The ablation season started at the end of April at the lower weather station (G490 at 490 m a.s.l.)
and in late-May at the higher station (G1060 at 1060 m a. s. l.) but terminated at the beginning of
October at both stations (Guðmundsson and others, 2002). In general the net-radiation made the
highest contribution to the total energy (Fig. 3-5 and Table 4-5), but was equalled or surpassed by
peaks in latent and sensible heat during occasional spells of high temperatures and strong winds
(Fig. 4a-b). Typically the daily variation in the energy budget was slightly more correlated to the
eddy fluxes than the net-radiation (Table 6, Fig. 6a-b and 7g-h). This correlation changed however
both with time and elevation during the summer (Fig. 6a-b and 7g-h). In June and July the net-
energy was mainly correlated to the net-radiation at the lower station (Fig. 3c-d, 4b-d and 7h),
where the albedo remained constantly low but the cloud cover was variable. At the higher station,
the cloud cover was generally lower and variations in the net-energy more related to the eddy
fluxes than the radiation (Fig. 7g).
Table 5. Observed summer balance ( bs ) and energy components on Hagafellsjökull over the ablation
season 2001. The parameters are explained in section 3.
                            Hl                      Hd                           R                      Mc                   bs
                           -2                       -2                       -2                      -2          -1
                   kW m          % of    kW m               % of      kW m           % of      kW m          m a of        m a-1 of
                                 Mc                         Mc                       Mc                       water         water
        G1060       0.79         8           2.12           21           7.14            71     10.05          2.60          2.36
        G490        2.89         11          8.62           31           16.18           58     27.69          7.35          7.35

Table 6. Correlation of calculated daily values of total energy ( M c from Eq. 3), with M m (Eq. 7), DDM1
from Eq. (8), DDM2 from Eq. (9), DDM3a from Eq. (10) with a constant ∆T ∆h , DDM3b from Eq. (10) with
the observed daily mean for ∆T ∆h , DDM4 from Eq. (11), temperature at the glacier ( TG ), temperature
north of Skjaldbreiður ( TS ), net radiation ( R ) and eddy fluxes ( H d + H l ).
             Mm       DDM1         DDM2         DDM3a              DDM3b         DDM4         TG          TS           R          Hd   + Hl
G1060         -        0.84           0.87           0.74           0.56          0.90        0.74        0.81        0.71            0.75
G490        0.95       0.74           0.80           0.80           0.80          0.81        0.66        0.77        0.68            0.74

        During the ablation season the net-radiation was controlled by the incoming short-wave
radiation and albedo (Fig. 4b-e, Fig 5c-h and Table 4) as the incoming long-wave radiation was
only slightly varying and outgoing long-wave radiation fairly constant at 315 W m-2 (Fig. 4d, 5e-f
and 6c-d). The time of exposure of the dirty low albedo summer surface (Guðmundsson and
others, 2002, p. 113), is evident in the radiation records (Fig. 4b-e and 5c-h). When the summer
surface was exposed on 11 June (Julian day 162) at G490, the albedo dropped to ~7%, and to
~37% on 18 August (day 230) in the firn at G1060. Because of the large drop in the albedo when
the summer surface is exposed, small winter balance tends to cause high summer melting (Fig. 8).
Similar trends have been found on Vatnajökull during the period 1996-2000 (Björnsson and others,
2001a, p. 13; 2001b; see Fig. 1 for location).
        At the beginning of the ablation season, in May and June, the melting at G1060 was mainly
kept up by the net-radiation (88-92% of the total energy; Fig. 5a and Table 4). Despite the high
albedo the net-radiation reached ~15-50 Wm-2 on average due to high sun elevation (Table 4 and
Fig. 5c,e,g). The temperature varied around 0 °C and the eddy fluxes were only ~0-6 Wm-2 on
average (Fig. 5c,i). Daily fluctuations in the energy budget were both due to variations in heat
fluxes and radiation (Fig. 6a and 7g).
        Cloud cover in the SW-Iceland was considerably below average in June 2001 and above
average in July 2001 (Jónsson, 2001). This is reflected in the monthly values of incoming short-
and long-wave radiations on the glacier (Fig. 5e-f). The highest net-radiation was measured during
July-August at G1060 and June-August at G490 (Fig. 5c-h and Table 4). During July-August, the
two warmest summer months, high eddy fluxes were also observed at both the AWSs (Fig. 5c-d,
5i-j and Table 4). At the lower station (G490), high temperatures and strong katabatic wind-flow
resulted in the eddy fluxes ( H d + H l ) contributing ~35-45% to the total-energy, almost equalling
the net-radiation supported by the high solar radiation ( Qi ) of ~150-165 Wm-2 and albedo of only
7% (Fig. 5b and Table 4).

                                                          Figure 5. Monthly averages of the energy budgets at
                                                          Stations G1060 and G490, compared with weather
                                                          parameters and albedo. The parameters are
                                                          explained in sections 2 and 3.

Figure 4. The energy components at G490 in
comparison with weather parameters and albedo,
displayed as three-day moving averages. (a):
Temperature ( T ) and wind speed ( u ). (b):
Energy components (identical to Figure 3d). (c):
The net radiation in (b), separated into short-wave
( Qi − Qo ) and long-wave ( I i − I o ) components.
(d): Incoming and outgoing short- and long-wave
radiation components. (e): Albedo.
                                                                                           components at
        September 2001 was remarkably warm Figure 6. Variation of the daily energyfor each month.
                                                   stations G1060 and G490, calculated
in SW-Iceland, ~2-2.5 °C above average (a-b): Standard deviation ( σ ) of daily values of the
(Jónsson, 2001). On the glacier the mean total-energy ( M ), net-radiation ( R ) and the eddy
temperatures were 4.9 °C and 1.1 °C at 490
                                                   fluxes ( H d + H l ). (c-d): Standard deviation of daily
and 1060 m a.s.l., respectively (Fig. 5i-j and
Table 4). As typical, stronger winds blew over values of the incoming ( Qi ) and outgoing ( Qo ) short-
Iceland in September than during the summer wave radiation and incoming ( I i ) and outgoing ( I o )
months of June-August (Fig. 4a, 5i-j and long-wave radiation.
Table 4; Guðmundsson and others, in
preparation). Hence, melt-rates were high due to the eddy fluxes despite the low solar radiation
(Fig. 3, 4b-d, 5a-f and Table 4). The eddy fluxes varied considerably duing September (Fig. 6a-b)
and they correlated strongly with the net-energy (Fig. 7g-h).

Figure 7. Comparison of physical and empirical models for estimating the glacier melt rate. The values in (a-
d) have been filtered by a three-day moving average. Left y-axis: melting. Right y-axis: energy supplied for
melting. (a-b): M c ≥ 0 (Eq. 3) compared to ablation calculated by Eqs. (8-9), using temperatures on and
outside the glacier, respectively. (c-d): Residuals between M c /( ρL1 ) ≥ 0 and the ablation estimated by Eqs.
(8) (Profile G) and (9) (Profile S). The mean values ( µ ) and standard deviations ( σ ) of the residuals are
also given, with subscripts referring to the two profiles G and S. (e-f): Correlation of M c ≥ 0 with the
ablation calculated by Eqs. (8) (Profile G) and (9) (Profile S). (g-h): Correlation of M c with net radiation ( R )
and eddy fluxes ( H = H d + H l ). The values in (e-h) are calculated for each day based on samples
extending from 15 days before to 15 days after (providing a moving window of 31 days). Values from Table
6 are shown as marks on the y-axis in (e-h).

4.2. Variation of the energy budget with elevation
A good agreement was found at Hagafellsjökull between observed summer balance ( bs ) and the
total energy ( M c ) calculated by Eq (3) (Fig. 9b and Table 5). The total ablation calculated from
the meteorological observations ( M c ) in G1060 was, however, slightly higher than that obtained
from the in situ stake measurements of the summer balance ( bs ). Similar discrepancy has been
observed in AWSs at higher elevations on Vatnajökull (unpublished data; see Fig. 1 for location)
where snowfall is frequent during the summer. Snow, which falls and melts during the summer, is
not detected by the measured total summer balance ( bs ) but the calculated M c includes energy
supplied for melting this snow.

Figure 8. Mean winter- ( bw ), summer- ( bs ) and
annual net-balance ( bn = bw + bs ) of the ablation
areas of Hagafellsjökull (excluding mass balance
data from the accumulation zone).

        All the energy components entering
the glacier increased downglacier (Fig. 9b and
Table 5). The long-wave radiation increased
due to higher cloud covers at lower elevations
on the glacier (Björnsson and others, 2000),
and the low albedo compensated for the
reduced global radiation (Fig. 9b-d). The
turbulent latent- and sensible heat fluxes were
                                                       Figure 9. Variation of the energy budget at Haga-
kept up by katabatic wind flows and the                fellsjökull over the entire ablation season 2001, as
increasing air temperatures downglacier (Fig.          related to elevation and compared with weather
9b,e). During the ablation season, solar               parameters, albedo and observed winter ( bw ) and
radiation heats up the low-albedo areas                summer ( bs ) balances. The y-axis remains the
outside the ice cap generating high lateral            same for every subplot. ELA and GT: altitude of
temperature gradients between the melting              the equilibrium line and the glacier terminus,
glacier surface and the surrounding areas (Fig.        respectively, in the year 2001. (a): Relative
9e and 10a-b), producing katabatic wind                contribution of the energy components. (b): Energy
downslopes       the     glacier    (Fig.     9e;      budget in comparison to the winter and summer
                                                       balances. Lower x-axis: power supplied for
Guðmundsson and others, 2002). The average             melting. Upper x-axis: corresponding water
wind speed was similar at both the AWSs on             equivalent units. (c): Means for Qi , Qo , I i and
Hagafellsjökull (~5-6 m s-1), however, slightly
increasing upglacier (Fig. 9e).                        I o . (d): Mean albedo. (e): Mean T and u . Lower
         In general, the relative contribution of      x-axis: T in °C. Upper x-axis: u in m s . The
                                                       parameters are explained in sections 2 and 3.
the energy components to melting was similar
for the two stations on the glacier: ~60-70%

Table 7. Degree-day factors of the models in Eqs. (8), (9) and (11), optimised by figuring in the entire
observations from the ablation season of 2001. i and ii: Parameters observed before (for snow) and after
(for ice/firn) exposure of the summer surface, respectively. The mean temperature difference is given
between both the Stations G1060 and S474 (G1060), and G490 and S474 (G490)
                                  ddf1           ddf 2           ddf 4          d
                                 mm °C -1
                                               mm °C   -1
                                                               mm °C   -1      °C
                               i      ii       i        ii      i        ii
                    G1060    10.6    11.3     7.9      9.3     5.3      6.0     -4.98
                    G490      9.1    11.1     9.9      10.2    6.3      8.1     -1.54

Figure 10. Variation of hour-mean values of air temperatures at Hagafellsjökull ( TG at G490 and G1060)
with temperature outside the glacier north of Skjaldbreiður ( TS at S474). (a-b): TG at G1060 and G490,
respectively. The scatter values in (b) are separated into southern and northern (katabatic) wind-flows in (c)
and (d), respectively. The models in Eqs. (12-13) are shown as imprinted grey lines in (a), (c) and (d). The x-
axis are the same for all the subplots.

from radiation and 30-40% from turbulent fluxes (Fig. 9a and Table 5). Upglacier in the
accumulation area the relative contribution of the net-radiation is expected to increase and there
radiation contributes to melting even though the eddy fluxes are negative (Fig. 9b,e). Such results
have been obtained in the accumulation zones of Vatnajökull (Björnsson and others, 2001a;

4.3. Empirical ablation models (degree-day)
The empirical models described by Eqs. (8-11) were applied for the ablation season of 2001 for the
two sites on Hagafellsjökull (G490 and G1060 at 490 or 1060 m a.s.l.), and the two AWSs outside
the glacier (S299 and S474 at 299 and 474 m a.s.l.). The optimised degree-day factors ( ddf 1 , ddf 2 ,
 ddf 4 ) and the mean temperature difference ( d ) of the ablation season 2001 are given in Table 7.
The ddf 3 parameter of Eq. (10) is not given, since the model predicted ablation poorly at the
higher station G1060. Higher ddf parameters were needed to describe the melting of ice/firn than
snow (Table 7). Guðmundsson and others (2003) found similar results for ddf 4 when applying the
model on Vatnajökull ice cap (see Fig. 1 for location), but in their case, lower values of ddf 1 and
 ddf 2 were needed to describe melting of ice/firn than snow.

4.3.1. Stability of the ddf parameters
        The empirical relationship between melting and temperature described by the ddf
parameters is typically assumed to depend entirely on conditions at the glacier surface and was
derived separately for snow and ice/firn surfaces. Assuming constant albedo (no surface changes)
during the period of June through August, the lower solar radiation and increased heat fluxes (Fig.
5c-f) resulted in reduced scaling factors (black lines in Fig. 11). This indicates that the degree-day
factors are sensitive to seasonal changes in the weather (time dependent), especially at the higher
station, G1060 (Fig. 11). The high ddf parameters in May (Fig. 11) are to be explained by the

Figure 11. Changes in ddf 2 (a,b) and ddf 4 (c,d) according to Eqs. (9) and (11), presented in relation to
albedo and to time for the locations G1060 and G490 Black lines: experimental monthly values of ddf
parameters obtained from the daily ablation calculated by Eq. (3) using observed weather data from 2001
and assumed constant albedo values (shown at the ends of the lines). Thick grey line: observed monthly
values of ddf 2 and ddf 4 in 2001, accompanied by the corresponding observed monthly albedo

relatively strong contribution of net radiation to melting (Fig. 11). As for the snow-ice transition,
its impact depends on its timing because of the seasonal variation in energy fluxes. For example, a
drop in the albedo at G1060 from 50% to 30% in June-July would increase ddf 2 from ~12 mm to
16 mm per °C (Fig.11a). In contrast, the same decline in albedo during July-August would actually
reduce ddf 2 . The parameter ddf 4 has a lower value, varies more gradually and is less sensitive to
changes in the weather parameters and to the timing of the snow-ice transition than ddf 2 (black
lines in Fig. 11); hence, ddf 4 comes nearer to depending solely on conditions at the glacier
surface. Furthermore, the typically strong contribution of net radiation in May, combined with the
strong winds and relatively warm temperatures of September 2001, significantly affected ddf 2 , but
not ddf 4 (Fig. 11a,c). Another justification for assuming time-independent scaling factors is that
the reduced solar radiation and increased heat fluxes as summer proceeds jointly counteract the
lowering of albedo (Fig. 5), which explains the slightness of developments in the observed
monthly values for ddf 2 and ddf 4 at both stations (grey lines in Fig. 11).

4.3.2. Degree-day models to predict the ablation 2001
        The predictions of the empirical models and the physical energy exchange model were
compared on a daily basis (Fig. 7a-f and Table 6). The degree-day models described the seasonal
variations in ablation at G490 and G1060 (Fig. 7a-b and Table 6) but did generally not predict
satisfactorily the daily values (Fig. 7a-d). However, the models produced some reasonable
predictions of the daily ablation at Hagafellsjökull during periods with high correlation between
the total energy and the eddy fluxes. Exceptions from this are seen in periods when the eddy fluxes
were more controlled by wind speed than temperature, e.g. in the middle of July 2001 at G1060

(Fig 7e,g). Better agreement was obtained between the observed daily ablation and predictions of
Eqs. (8, 9 and 11) at the higher (G1060) than at the lower (G490) station (Fig. 7a-f and DDM1,
DDM2 and DDM4 in Table 6). The best predictions were obtained at both the stations applying
Eqs. (9) and (11) that uses temperature observations outside ( TS ) rather than inside ( TG ) the
glacier (Fig. 7e-f and Table 6). Equation (11) that uses the constant lapse rate ( γ ), gave slightly
better results than Eq. (9) that uses the absolute temperature difference d (DDM2 and DDM4 in
Table 6). Guðmundsson and others (2003) obtained similar results of using Eq. (11) on
Vatnajökull ice cap, but the performance model in Eq. (9) was much poorer than in our case for
Langjökull. It should though be noted, that due to the damping effects off the melting glacier, the
TS projected to a higher elevation with a constant lapserate, is neither a resample of the absolute
values or variations of the temperature at the same elevation within the boundary layer of the
glacier (Figs. 9e and 10a,b). The better performance of using TS rather than TG was particularly
evident at the lower station. Further, from the end of May through July the melting was mainly
driven by the incoming solar-radiation ( Qi ) (Fig. 4 and 7h), which was better reflected by TS than
TG .
        The models that uses temperature form outside the glacier (Eqs. 9-11) did almost equally
well in estimating the daily ablation at the lower station. In contrast, Eq. (10), that uses
temperature gradient between two stations from outside the glacier, failed at the higher station
(Table 6), particularly when observed daily means of ∆T ∆h were used instead of constant value.
The standard deviation of the residuals between daily values of the observed and calculated
ablation at the higher station with Eq. (10) was σ L 1 = 13 mm d-1 of water using a constant value,
and σ L 2 = 25 mm d-1 of water when applying daily values of ∆T ∆h . This is much higher than
σ G = σ S = 5 mm d-1 of water obtained when using Eqs. (8), (9) and (11) (Fig. 7c). The poor
performance of Eq. (10) is partly explained with a very low correlation and negative that was
found between hourly mean values of the temperature difference between the glacier and the
surrounding non glacierized areas ( d ), and the temperature gradient ∆T ∆h observed between
two stations outside the glacier, or –0.29 and –0.18 at G490 and G1060, respectively.

4.4. Sensitivity of the summer melting to climate changes
Both the energy balance model and the degree-day models have been used to study the glacier
mass balance response to a prescribed climate change. This was done by calculating the sensitivity
of the melting rate and the equilibrium line altitude (ELA) at Hagafellsjökull to given regional
temperature changes ( ∆TS ) outside the ice cap (at S474). An empirical relationship between the
changes in air temperature outside the glacier and the temperature and katabatic wind speed over
the glacier were included in the calculations of the eddy fluxes.

4.4.1. Empirical relationships between temperature inside and outside the glacier and
katabatic wind
          Piecewise-linear regression of the data in Fig. 10, relating the temperature T (z ) = TG at the
height z = 2 m above the melting ice surface at the two sites G490 and G1060, to the temperature
( TS ) at S474 gave the relationships

       TS − 4.51;    TS < 4.51
TG =                                 ,                                                              (13)
       0.53TS − 2.38;    TS ≥ 4.51

       TS − 0.37;    TS < 3.67 or 85° ≤ WD ≤ 275°
TG =                                                                       ,                      (14)
       0.28TS + 2.28;    TS ≥ 3.67 and      (WD ≥ 275° or WD ≤ 85°)

for G1060 and G490, respectively, where WD is the observed wind direction (Fig. 2). The
temperature TS does not reflect the variations of the temperature TG used in the energy budget
calculations (Eqs. 4-5), and hence, it is not preferable to use Eqs. (13-14) to calculate TG as a
function of TS . Instead, the equations were used to quantify possible variation of TG ( ∆TG ) due to
a given global temperature changes ( ∆TS ) (Figs. 12a and 13a).
        Katabatic wind flow is important when calculating the energy balance (Eqs. 4-5). The
katabatic wind down the glacier outlet is expected to be driven by the temperature gradients
between the glacier and the surroundings of the ice cap (Figs. 9e and 14a), similar to that observed
on Vatnajökull (Guðmundsson and others, 2003; see Fig. 1 for location). Due to the damping
effects of the melting glacier, the temperature gradient between the glacier and its surrounding
areas changes with changed global temperature (Fig. 14a). We adopt the linear expression

uV =       TV + c = 0.94TV + 3.67 ,                                                               (15)

which was derived for data on the summer means of uV (in m s-1) and TV , observed at AWSs at
1200 and 1100 m a.s.l. at the northeasten and western Vatnajökull, respectively, 1994-2001 (data
from Björnsson and others, 2001a; 2001b and Guðmundsson and others, 2003). We use the ratio
∆u / ∆TG = ∆uV / ∆TV = 0.94 to obtain a plausible change in the wind-speed ( ∆u ) with ∆TG (or
∆TS ) at the two sites on Hagafellsjökull (Figs. 12b and 13b).

4.4.2. The energy balance for scenarios of climate changes
        The sensitivity of the glacier mass balance to climate changes was studied by computing
the eddy fluxes at the two stations on Hagafellsjökull, for temperature variations ( ∆TS ) of –5 to 5
°C from the observed values of TS in 2001. Changes in the albedo (net-radiation) are indirectly
integrated because the winter balance ( bw ) was assumed to be equal to that observed in 2001 and
by melting the winter snow the mean is significantly altered. The observed solar radiation ( Qi ),
incoming ( I i ) and outgoing ( I o ) heat radiation were assumed to be equal to that observed in 2001.
The results (Fig. 12-14) are shown as averages over a fixed ablation period close to that of 2001.
The various energy components (Fig. 12e-f and 13e-f) were calculated for each ∆TS both by
assuming constant wind speed (the same as observed in 2001) and the wind speed depending on
TS (Fig. 12b and 13b).
        Assuming I i and bw to be constant may underestimates R and the total energy ( M c )
when ∆TS >0 °C, and vice versa if ∆TS < 0 °C. Increased regional temperature would reduce bw
(as long as winter accumulation does not increase), lower the summer mean albedo increasing the
global radiation absorbed by the glacier and extend the ablation season. Assuming constant I i
(rather than proportional to TS4 ) leads to at most ~10% and ~30% errors in the total energy at the
lower (G490) and higher (G1060) station, respectively, for | ∆TS | ≤ 3 °C. The assumption of
constant bw , was cautiously estimated to cause at the most an error in M c of ~10-15% and ~20%

Figure 12. Weather parameters and energy components at G1060 as functions of assumed regional
temperature changes at S474, as well as comparisons of the total energy balance components provided for
melting with predictions from the empirical degree-day models in Eqs. (8), (9) and (11). All parameters
represent averages over a period equal to the observation season in 2001. Lower x-axis: temperature at
S474 ( TS ). Upper x-axis: changes ( ∆TS ) from TS in 2001 (a): Temperature ( TG ). (b): Wind speed. (c-d):
Relative contribution of the energy components in (e) and (f), respectively. (e-f): Energy budget using only
the temperature changes in (a) and the combination of the temperature and wind-speed changes in (a) and
(b), respectively. Left y-axis: sum of power supplied for melting. Right y-axis: water equivalent units. (g-h):
 M c from (e) and (f), respectively, compared to the melting rate a s from Eqs. (8), (9) and (11). Left y-axis:
melting. Right y-axis: corresponding power needed for the melting. (i): Albedo. (j): Day of the year (left y-
axis) and the corresponding month (right y-axis) when surface melting reaches the summer surface from the
previous year. Black profiles are calculated using only the temperature changes in (a) and light-grey profiles
using the combination of temperature and wind-speed changes in (a-b). The summer surface does not
become exposed at G1060 when ∆TS ≤ −2 °C.

Figure 13. The variation in weather parameters and energy components at G490 that would accompany
assumed regional temperature fluctuations at S474, and comparisons of the total energy balance
components provided for melting with predictions from the empirical degree-day models in Eqs. (8), (9) and
(11). All parameters represent averages over a period equal to the observation season in 2001. See the
caption with Fig. 12 for explanations of the subplots.

at G490 and G1060, respectively. This error would be less than 5% when -3 °C ≤ ∆TS ≤ 5 °C at
G490 and negligible at G1060 when ∆TS ≤ −2 °C because the summer surface would not be
exposed (Fig. 12j).
        Increased melting with rising global temperature ( TS ) would be due to increased eddy
fluxes rather than higher net-radiation (Fig. 12c-f and 13c-f), albeit earlier exposure of the summer
surface and subsequently reduced albedo (Fig. 12i-j, 13i-j and 14c). An increase of the global
temperature by 3 °C and assuming constant wind speed would increase melting by ~60% (from
~2.6 to ~4.3 m a-1 of water) at 1060 m a.s.l. and ~30% (from ~7.2 to ~9.4 m a-1 of water) at 490 m

a.s.l. (Fig. 12e, 13e and 14b). Assuming katabatic wind to increase proportional to TG , ablation
would increase by ~90% (from ~2.6 to ~5.1 m a-1 of water) at 1060 m a.s.l. and ~70% (from ~7.2
to ~12.5 m a-1 of water) at 490 m a.s.l. (Fig. 12f, 13f and 14b). Lowering the air temperature by 3
°C, the reduction in melting would be similar whether we account for changes in wind speed or not
(Fig. 12e-f, 13e-f and 14b); ~50% and ~60-70% at 1060 and 490 m a.s.l., respectively,
         Predicted changes in the equilibrium line altitude (ELA) with ∆TS are shown in Fig. 14d.
The ELA on Hagafellsjökull was observed at ~1140 m a.s.l. in 2001 ( ∆TS = 0 °C) but is estimated
at ~1000 m and ~1300 m a.s.l. for ∆TS of –3 °C and 3 °C, respectively. Assuming constant bw we
may underestimate the rise in ELA for ∆TS > 0 °C, and underestimate the descent subsequent to
cooling. The summer balance bs (or M c ) varies non-linearly with elevation (Fig 14b). The
bending of the ELA profile for extreme values of ∆TS (grey circles in Fig. 14d), reflects the
impact of the wind-driven eddy fluxes. For extreme cooling they become a sink of energy.

4.4.3. Comparison of the physical and empirical models for temperature changes
        A performance of the degree-day models was tested by using a comparison with the results
from the physical energy balance model. The degree-day models are more reliable at the higher
than the lower stations (Fig. 12g-h and 13g-h). Reasonable agreement was obtained between the
complete model of physical energy balance and equations (8), (9) and (11) at G1060 when –3
°C ≤ ∆TS ≤ 3 °C (Fig. 12g-h), and this also applied to G490 for –4 °C ≤ ∆TS ≤ 2 °C, assuming a
wind speed proportional to temperature (Fig. 13h). At the higher station, the models in Eqs. (8) and
(11) are closer than Eq. (10) in simulating the influence of the assumed wind speed changes (Fig.
12h). The empirical models diverged increasingly from the physical ones as the temperature
departed from the reference temperatures of 2001 (Figs.12g-h and 13g-h), especially at the lower
warmer station (G490).

Figure 14. Predicted changes in meteorological parameters and ELA due to a prescribed deviation ( ∆TS )
from the mean summer temperature of 2001 at S474, as a function of elevation. The parameters represent
averages over a period equal to the observation season in 2001. The y-axis stays the same for all the
subplots. ELA: equilibrium line altitude in 2001. GT: altitude of the glacier terminus. The results in (b-d) are
obtained by assuming temperature changes only (black lines) and by assuming changes in both
temperature and wind speed (grey lines). (a): Changes in air temperature. (b): Changes in the energy
balance (summer balance). Lower x-axis: power supplied for melting. Upper x-axis: the water equivalent.
Here bs is the observed summer balance in 2001. (c): Albedo values, (d): ELA. Lower x-axis: mean values
of TS during the observation season. Upper x-axis: corresponding changes ( ∆TS ) of TS . The shifts in ELA
were derived using the winter balance at the ELA of 2001, i.e.   b w = − bs   as seen in (b), and a linear extension of the
inferred values for   Mc   in (b).

Energy balance studies described the summer ablation 2001 successfully. This applied to temporal
and spatial variations. In the beginning of the ablation season the temperature was close to the
melting point, the albedo high and the energy components were small and the eddy fluxes even
negative. Some ablation took place through radiation even at air temperatures below the melting
point. During the warmer summer months (June/July-August) ablation was both due to high eddy
fluxes and net-radiation, that contributed ~50-70% to the total energy. September was unusually
warm and strong autumn winds produced high eddy fluxes and considerable melting in spite of
low global radiation. This would describe conditions of warmer autumns and winter months. In
general, the input of energy from all components was highest in the terminus due low albedo and
high temperature. The relative contribution of the net-radiation to the total input was however
higher in the accumulation area, albeit higher albedo in the ablation area; increasing from 60% at
the terminus to 70% at the equilibrium line and frequently 100% on the highest parts of the glacier.
Temporal variations in the melting, however, were more due to fluctuations in the eddy fluxes than
radiation, at all elevations.
      Seasonally changes in the melting were reasonably described with empirical degree-day
models. A linear extrapolation of the mean temperature outside the glacier is not a satisfactory
estimate of the mean temperature on the glacier. Temperature gradients outside and inside the
glacier differ. A regression model applying the temperature observations outside rather than inside
the glacier did best describing the ablation. This was most obvious when the melting mainly varied
with global radiation, which contribution was better described by air temperature at the low albedo
surroundings while this information tends to be smoothed out in the temperature signals over the
melting glacier. The optimised regression coefficients derived for the climate conditions in 2001
cannot be expected to apply in general. They may vary with the relative contribution of the
radiation and eddy flux components. The turbulent fluxes are influenced by the temperature
difference between the glacier and the surrounding areas, which drive katabatic wind. Further
measurements would be required to gain experience of how well degree-day models may describe
seasonal changes in ablation for various glacier surface characteristics and the melting for given
scenarios of climate changes.
      Global warming would lead to extended ablation season in the spring and the autumn, even
increased ablation during the winter. Enhanced melting may lead to earlier exposure of the low
albedo summer surface and increased net-radiation. In that respect the winter balance is important
for the mean albedo of the subsequent summer and the summer melting. The turbulent fluxes
would increase. Increased temperature difference between the glacier and the surroundings would
generate stronger katabatic wind downglacier. Application of the energy balance model for studies
of the melting for given scenarios of climate changes would require continued work on the
relationship between global warming and the katabatic flow.

This work was supported by the National Power Company of Iceland, the Nordic project Climate,
Water and Energy (CWE), The University of Icleand Research Fund and the EU projects Icemass
(ENV4-CT97-0490) and Spice (EVK2-CT-2002-00152).

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