Jökull - 01.01.2009, Page 5

Langjökull, energy balance and degree-day models
Hd = !1cpk20u(z)
T (z)" T (zT )
(ln(z/z0) + # zL)(ln(z/zT ) + #
z
L )
(6)
and
Hl =L"k20u(z)(0.622
!1
P
)·
e(z) " e(zQ)
(ln(z/z0) + # zL)(ln(z/zQ) + #
z
L)
(7)
where T (z), u(z), and e(z) are the air temperature
in !C, the wind speed in m s"1 and the vapour pres-
sure in Pa, respectively, at a height z above the sur-
face. The roughness length for velocity (z0) is de-
fined as the height at which the wind speed is zero,
and for temperature (zT ) and water vapour (zQ) as
the heights at which the semi-logarithmic temperature
and water vapour profiles extrapolate to the surface
value. Once z0 is known, zT and zQ were estimated as
suggested by Andreas (1987). On a melting surface,
T (zT ) $ 0!C and e(zQ) $ 611.213 Pa (e. g. Oke,
1987). The parameter k0 = 0.4 is the von Kármán
constant, cp = 1010 J kg"1 K"1 is the specific heat
capacity of air under constant pressure,L" = 2.5 ·106
J kg"1 is the specific latent heat of evaporation. The
density of the air is included as !1 = !0(P/P0), in
which !0 = 1.29 kg m"3, P0 = 1.013 · 105 Pa, and
P is the air pressure in Pa, estimated with Eq. 1. The
Monin-Obukhov length (e. g. Munro, 1989; Björns-
son, 1972) is expressed for one-level measurements
and when z >> z0 as
L = "A + 1
B
(8)
assuming A = #z/(ln(z) " ln(z0)) and B =
(g/T0)(T (z)/u2(z))(ln(z) " ln(z0)), where g =
9.8 m s"2 is the acceleration of gravity. Published
values for the empirical stability correction constant
# are typically !5 to 8 (e. g. Dyer, 1974; Högström,
1988, 1996), but variations within this range only
cause a small uncertainty in the calculated sensible
heat flux (Munro, 1989). This is supported by our
energy balance calculations that show the same to be
relevant for the latent heat flux. Here, # = 7 was
chosen as a practical approximation for both Hd and
Hl.
We assume that the z0-values in Table 2, constant
with time and only varying with the surface type, are
appropriate for the presented study. The values are
in a close agreement with more accurately estimated
z0-values of Brock et al. (2006), derived by microto-
pographic and wind profile measurements at the Haut
Glacier d’Arolla, Switzerland. Typically z0 is in the
range of 1–10mm for glacier firn and ice (e. g. Björns-
son, 1972; Moore, 1983; Morris, 1989; Greuell and
Konzelmann, 1994; Braithwaite, 1995b; Hock and
Holmgren, 1996; Brock et al., 2006), but values up
to 7–10 cm have been reported for the rough lower-
most ablation areas of Vatnajökull ice cap in Iceland
(Smeets et al., 1999). Generally, the temporal vari-
ation of z0 during the ablation season is unclear (e.
g. Brock et al., 2006), but Denby and Smeets (2000)
did not record any variations in z0 for ice over several
months on southern Vatnajökull.
Table 2: Applied values of surface roughness (z0). –
Hrjúfleikastuðull jökulyfirborðs.
z0 ln(z0)
mm
New snow ! 0.1 -9.2
Melting snow/firn ! 2 -6.2
Ice in ablation zone ! 10 -4.6
The selected z0-values gave in general a good fit
between the derived values of Mm (Eq. 3) and Mc
(Eq. 4). Varying z0 from 1 to 14 mm in our calcula-
tions alters the total melting energy at most by 3% at
G1100, and by 7% at G500 when changing z 0 from
1 to 7 cm. The high consistency between the derived
Mc andMm values (e. g. Figure 2b) indicates that the
sonic echo sounder satisfactorily describes the cumu-
lative daily melting rates despite the rather high un-
certainty of the sonic echo sounder (Table 1). Up to
95% of the daily variation inMm is described byMc,
and the standard deviation of the difference between
daily values of Mc and Mm is 33 and 20 W m"2 at
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