Jökull - 01.12.1972, Page 48

where
Ly is the latent specific evaporation heat
m' is the fluctuation of the specific hu-
miclity
The eddy-flux density of vapor is given by
e=“l (?)
Lv
Phase changes are assumed only to take place
on the surface and tlie vertical divergence in
the radiation fluxes is neglected.
Under the assumption of homogeneous uni-
formity and stationarity in the Prancftl layer
these three fluxes are constant with the height
(Calder 1966). The fluxes could therefore be
measured clirectly in an arbitrary height in the
Prandtl layer, but as this requires very com-
plicated instruments one prefers to express the
fluxes as functions of the vertical mean gradient
fields which are less difficult to measure. The
procedure is based on the similarity theory and
tlie semiempirical theory of turbulence. The
fluxes can then be written as
lim u (z) = 0 (11)

lim T (z) = : 0 °C (12)

lim e (z) = 6.11 mb (18)
Z—> z"o
where zo, z'o ancl z"o are the integration con-
stants for the wind—, air temperature—, and the
vapor pressure profile respectively.
The greatest difficulty in solving the system
of equations (8), (9) and (10) is involved in
obtaining the diffusivity coefficients KM, KH,
Kw expressed with the mean fields of wind,
air temperature and humidity. No complete
theory exists. In the glaciological literature
some authors have contributed with semi-
empirical solutions (Sverdrup 1936, Wallén
1948, Hoi.nkes and Untersteiner 1952). In the
present paper a solution based on the Monin
and Obukhov (1954) similarity theory is ap-
pliecl. According to this theory the three co-
efficients can be written
du
T = p dz = p Ux“ (8)
Hd = p kh / dT g \
(dT + v) (9)
p ux Tx
L„
K„
dm
dz
p ux mx
(10)
where usual symbol convention is adopted. All
variables are averages but the bar has been
dropped for convenience in printing. The left
side of the equation can be considered as a
definition of the turbulent eddy diffusivity co-
efficients, KM, KH and Kw. The right hand
result is a result of the Monin and Obukhov
(1954) theory. The scaling parameters ux, Tx
and mx for respectively wind velocity, air
temperature and the specific humidity are con-
stants in the Prandtl layer. Following the mete-
orological convention the fluxes are defined
positive if directed upwards.
For a melting glacier equations (8), (9), (10)
are solved with the boundary conditions
km =
K Ux Z
cpM
KH =
K UX Z
<Ph
Kv
K ux Z
(14)
cpw
where q>M, cpn an<i cpw are functions of the
stability parameter z/L. The Monin-Obukhov
scaling length is given as
L =
T0
uxz
K Tx
(!5)
where g is the acceleration of gravity ancl To
(constant) is the mean potential temperature
in the Prandtl layer.
By assuming that the mechanism of the eddy
transfer of momentum, sensible and latent lieat
is of such nature that
Ku =
K„
(16)
the problem is further simplified. Then the
three coefficients and the mean profiles for
wincl speed, air temperature and humidity can
46 JÖKULL 22. ÁR