J.
Fluid
Mech.
(1980), vol.
100,
pa~t
2,
pp.
367-384
Printed in
&eat
Britain
367
Ice blocks melting into a salinity gradient
By
HERBERT E. HUPPERT
Department
of
Applied Mathematics and Theoretical Physics,
University
of
Cambridge
AND
J.
STEWART TURNER
Research School
of
Earth Sciences,
Australian National University, Canberra
(Received 21 June
1979
and in revised
form
21 November
1979)
In
our
previous qualitative paper, it was shown that when a vertical ice surface melts
into a stable salinity gradient, the melt water spreads out into the interior in a series
of nearly horizontal layers. The experiments reported here are aimed at quantifying
this effect, which could be of some importance in the application to melting icebergs.
Experiments have also been carried out with heated and cooled vertical walls at
larger Rayleigh numbers
R
than those
of
previous experiments.
The main result is that for most of
our
experiments there is no significant difference
between these three cases when properly scaled. The layer thickness over a wide range
of
R
is described to within the experimental accuracy by
where the term in brackets is the horizontal buoyancy difference evaluated at the
mean salinity and
dpldz
is
the vertical density gradient due to salinity. In the case
of
ice melting into warm water the effective wall temperature
T,
is
approximately
0
"C,
whereas in colder water the freezing point depression must be taken explicitly into
account. A detailed examination of the vertically flowing inner melt water layer in
both homogeneous and salinity stratified cases has been made. This layer and the
melt water which
is
mixed outwards from
it
into the turbulent horizontal layers have
little effect
on
the outer flow. At high
R
and large external salinity, however, mixing
can reduce the effective salinity at the inner edge
of
the horizontal layers, and thus the
layer scale. A puzzling feature
is
the relatively weak dependence of layer scale on local
salinity, though the vigour of convection and the rate of melting are greater where the
salinity is high.
The direct application
of
our
results to oceanographic situations predicts layer
scales under typical summer conditions of order tens of metres in the Antarctic and of
order metres in the Arctic. More measurements will
be
needed, especially close to
icebergs, before the application of these ideas to polar regions can be properly evalua-
ted.
0022-1 120/80/4553-2
140
$02.00
0
1980
Cambridge
University
Press
368
H.
Id.
Huppert
and
J.
X.
Turner
1.
Introduction
Each year approximately
5000
icebergs are calved from Antarctic glaciers and
float into the adjoining seas. These icebergs initially have horizontal sizes of typically
1
km and
a
depth of typically
250
m, the mean thickness of the glacier
or
ice shelf
at the point of calving. Arctic icebergs are much more numerous, though smaller.
Approximately
25000
are calved each year, with a typical horizontal scale of
250
m
and a typical depth of
50
m.
The work described in this paper is motivated by the
melting of these icebergs, and the specific question examined is this: what happens
when
a
vertical wall of ice melts into stratified water?
Melting can take place on the top, on the bottom and along the sides of an iceberg.
We arque that melting from the sides is likely to exceed that from either the top or the
bottom. This
is
because in polar regions the air around the iceberg is generally at
a
temperature below tlie melting point of the ice. Also the top of an iceberg is usually
covered by
a
layer of slush, which acts as an insulator and inhibits further melting. At
the bottom, the melt water produced is lighter than its surroundings, but is restrained
in its rise because the iceberg is in the way
!
Thus new ambient fluid, which is required
to supply heat for further melting, is restricted from coming into contact with the
iceberg. This is not
so
for melt water produced at the side, which is not constrained in
its movement, and the melt process is able
to
continue efficiently in the manner to be
described below.
The simplest description
of
the melting process would be as follows. Melt water is
lighter than its surroundings because it,s density deficiency due to the salinity differ-
price exceeds the density excess due to its lower temperature. Being lighter,
it
rises up
the sides
of
the iceberg in
a
thin boundary layer uncontaminated by the surrounding
salty
water and is deposited at the surface. Once at the surfave the water
would
spread
1iorizont;Llly
as
a
gravitjr vrirrent if nnimpeded, or could be syphonetl off as desired.
‘Phis latter concept
was
considered
by
a
number of groups throughout the world who
:ire
currently investigating the harvesting of icebergs for their fresh water.
A
thin nomentraining houndnry layer will only occur
if
the Reynolds number, Re, is
no
greater than
about
103.
ITe can estimate the Reynolds number by using the laminar
solutions for flow induced by heating a vertical plate (Chapman
1960,
ch.
9)
and
:bssiiming that the density difference
Ap
driving the flow is due principally to either
the temperature difference or the salinity difference
-
in the polar case,
it
is decidedly
the latter. The resnlt is that the velocity at the top of the boundary layer is approxi-
mately
(I.qAp/po)i,
where
L
is the length of the side
wall,
g
is the acceleration due to
gravity and
po
is the mean density. The criterion for
a
non-entraining boundary layer
is thus that
Re
=
(L3gAp/po):v-l,
where
v
is the kinematic viscosity, be no greater than
about
t03.t
With
Ap/p,
=
10-3,
L
=
102
m and
v
=
1
0-6
m2
s-l,
reasonable values for
icebergs in polar seas,
Re
-
108,
very much greater than
lo3.
For icebergs in mid-
latitude seas, the density difference is up to an order of magnitude larger.
The
suggestion that melt water rises in a turbulent and hence entraining boundary
layer
was
recently put forward by Neshyba
(1977).
He modelled the ocean by
a
50
m,
t
Convection driven by an imposed density difference across
a
vertical wall
1s
often analysed
In terms
of
the Crashof number
G
=
gApL3/p0
v2
=
Re2.
The laminar convection
solution
13
experimentally observed to be
a
correct descriptlon
of
the flow pattern only for Grashof numbers
smaller than approximately
lo6.
Ice
blocks melting into
a
snlinity gradient
369
-2
-1
T(T)
0
1
-2
-I
T("C)
0
1
h
5
200
-
9
(b)
400
34 4 34
5
34
6
34
1
34 4 34
5
34
6
34
7
S
(%a
)
S(%O)
BIL~
t<h
1.
'l'wnlwrature
(7')
and salinity
(S)
profiles measured in the upper
400
m
of
the \Vedclcll
Scd
(at1,tf)ted
froin
figure
3
of
Foster
B
Carmack
1976).
(a)
NeaI
the Scotia
JAdge
at the northern
ctlgr
of
thc
wit.
(0)
Near the turnlng point
of
the curient gyre.
PIGUKE
2.
Typical mean salinity profiles in the Arctic (Hellerman, private communication).
(1)
772'
N,
763"
W,
summer.
(2)
67;'
N,
60;"
\V,
summer.
(3)
6'36"
N,
6iF
W,
autumn.
relatively light layer above a deep, heavy layer
of
greater salinity. Neshyba then
assumed that the amount of entrainment
is
determined by the criterion that the
mixed fluid at the top of the boundary layer has the same density
as
the surface
water. Using this criterion, he calculated that, the entrainment ratio, the ratio
of
entrained water to melt water, for Antarctic icebergs is approximately
60.
This
implies that the total upwelling around Antarctic icebergs is of the order
of
1
0l1
m3
per day, which represents
a
considerable vertical transport.
The
aim
of
this paper
is
to
show
that another, and
in
our opinion more likely,
mechanism pan occur. The essential feature
of
this mechanism is that
it
takes explicitly
into account a variation
of
salinity with depth. Non-uniform vertical salinity distri-
butions, either smooth or containing steps, are typical in most parts of both the
370
H.
E.
Huppert and
J.
S.
Turner
Antarctic and Arctic. Some observed profiles are plotted in figures
1
and
2.
We shall
demonstrate that the inclusion of a salinity gradient leads to a completely different
flow pattern to either of the ones described above. The flow described
is
shown in figures
7
and
8
(plate
2)
and figure 9 (plate
3).
Next to the ice there is a thin inner boundary layer
of rising cold melt water. The water in this boundary layer mixes with that in
a
thick
outer boundary layer, in which the water moves
downwards.
Because the density of
the ambient fluid increases with depth, water in the outer boundary layer can only
descend until
it
reaches the level where its density equals that of the far-field water.
At that point, the water commences to move horizontally. The overall pattern outside
the boundary layers is a series of nearly horizontally oriented layers, each layer being
separated by a relatively thin interface from the adjoining layer. The water from the
melting iceberg is mainly deposited in these layers.
A
qualitative description of this process appeared in
Nature
(Huppert
&
Turner
1978); the purpose of the present paper is to discuss the quantitative details.
Our
presentation of the discussion will be given in three stages. In the next section we
discuss experimental and analytical results for heating
or
cooling a vertical salinity
gradient from the side (without allowing for the introduction of melt
or
any other form
of
water). In
§
3
we consider the melting of ice in a fluid of uniform salinity. In
§
4
we
analyse the effects of a salinity gradient on the melting process. In
$5
we discuss
pertinent oceanographic data and their relationship to
our
experiments; and
our
conclusions are summarized in
9
6.
2.
Horizontal temperature differences
Some aspects of heating a stable salinity gradient from the side have been considered
previously by Thorpe, Hutt
&
Soulsby (1969), Chen, Briggs
&
Wirtz (1971), Wirtz,
Briggs
&
Chen (1972) and Chen (1
974).t
These investigations were primarily concerned
with the onset of motion
at
low Rayleigh numbers. The work of Thorpe
et
al.
was
further restricted in that most of the experiments and all
of
the theory were carried
out
for
the case when the fluid is confined by a narrow slot. The width
of
the slot then
sets both the vertical and horizontal length scale of the motion. The experiments
of
Chen
et al.
(1971) indicated that there is a critical value of the Rayleigh number, to be
defined in (2.2), below which the only motion
is
a
very slight rise of warm fluid near the
heated wall. Above the critical Reyleigh number, the viscous forces are overcome and
warm fluid parcels near the heated wall rise. Because the diffusion coefficient of salt
is
so
much smaller than that
of
heat, such parcels retain almost all of their salt as they
rise in
a
continually decreasing external density field. Thus their rise can be no more
than
7
=
rP(T-9
8,)
--P(T,,&)I
Z’
(2.1)
IdP
where
p
is the density as a function of temperature and salinity, with
00
and
w
repre-
senting values in the far field and at the wall, respectively. After this rise the fluid must
turn into the interior. Motion thus takes place in a series of layers as depicted in
figure
3
(plate
l),
taken during one of
our
experiments to be described below. The
t
The
flow
produced by heating a horizontal cylinder in
a
stable salinity gradient
is
discussed
by Hubbell
&
Gebhart
(1974).
Ice blocks melting into
a
salinity gradient
371
thickness of the layers scales with
7.
The flow
is
towards the heated wall at the bottom
of each layer and away from the wall along the top. The layers slope gently upwards
towards the wall because at each interface inflowing fluid lies above outflowing fluid
which
is
relatively hotter (due to the heat transfer from the wall) and saltier (due to its
deeper origin). This is exactly one of the situations for which double-diffusive convec-
tion occurs (Turner
1973,
cha.
8)
and heat is transferred upwards across the interface in
preference to salt. The inflowing fluid hence decreases in density as
it
progresses to-
wards the wall and has an upward component of motion. Using
7
as the vertical length
to evaluate the Rayleigh number
where
KT
is
the molecular diffusivity of heat, and using the (vertical) mean value of
S,,
Chen
et
al.
determined from their experiments that the critical value of the Rayleigh
number,
R,,
is
15
000
f
2500.
Analysing the results of the nine experiments ranging in
Rayleigh number from
14000
to
54000
for which layers occurred, they conclude that
the thickness of the layers,
h,
expressed as a fraction of
4,
is
0.81
0.10,
with no syste-
matic dependence on the Rayleigh number.
In a number of different numerical calculations with
R
=
lo5,
Wirtz
et al.
(1972)
and
Chen
(1974)
obtained results in broad agreement with the experimental results.
Since the Rayleigh number calculated via
(2.2)
of
an iceberg in polar seas is about
it
is of interest to determine the form of flow and layer scale for Rayleigh numbers
very much larger than critical. We thus conducted the following experiments.
For
the
first of two series of experiments we centred a thin-walled, stainless steel circular
cylinder of radius
44
cm and height
33
cm (a photographic developing cylinder, to be
explicit) in a rectangular Perspex tank
40
x
20
x
30
cm high. The space between the
cylinder and the container was filled with water at room temperature, linearly strati-
fied with salt, such that there was fresh water at the surface. The filling was accom-
plished using the standard double-bucket procedure first suggested by Oster
&
Yamamoto
(1
963)
and Oster
(1
965).
Hot water was then introduced into the cylinder
and
its
temperature held constant by the use of a temperature controlled heating
coil, known as a Thermoboy. Motions in the water were made visible by using a
shadowgraph technique. The layer thicknesses were monitored during the experiments
and frequent photographs were taken
so
that details of the experiments could be
reconsidered later and analysed at leisure.
For
the second series of experiments, a thin-walled aluminium box
8
x
20
x
40
cm
high was placed at one end of a rectangular Perspex tank
50
x
20
x
40
cm high. The
procedures of the first series of experiments were followed except that for most of the
experiments the surface was quite salty. The second series was carried out entirely to
determine whether the salinity
at
the surface played any role
(as
might be concluded
from the experiments with melting ice, to be discussed in
5
4).
The external parameters
and results of all the experiments are presented in table
1.
The non-dimensional layer
thicknesses,
h/y,
are graphed in figure
4
as a function
of
the Rayleigh number calcu-
lated as in
(2.2)
with
K~
and
v
evaluated at the wall temperature and zero salinity. We
note here that we have used the mean far-field salinity in evaluating
7.
The variation of
7 with salinity and the appropriateness of using the mean value will be discussed in
3
5.