Introduction
Over the past 40 years, Arctic summer sea ice extent has reduced by 50 %,
making it one of the most sensitive indicators of man-made climate change
. This rapid decrease is
at least partially due to the ice-albedo feedback
. Moreover, if the
ice-albedo feedback is strong enough it could lead to instabilities and
abrupt changes in ice coverage in the future
. The albedo of ice is
significantly reduced by the presence of melt ponds on its surface
. Therefore,
understanding the evolution of melt ponds is essential for understanding the
ice-albedo feedback and, consequently, the evolution of Arctic sea ice cover
in a warming world. This means that accurate melt pond parameterizations must
be incorporated into global climate models (GCMs) to improve their sea ice
forecasts . The main difficulties
with including accurate melt pond parameterizations in large-scale models are
that pond evolution is nonlinear and that it is the result of a variety of
different physical processes operating on a range of length and time scales.
For these reasons, it is important to understand the mechanisms that drive
the evolution of melt ponds.
Typically, the evolution of pond coverage on first-year ice proceeds in
fairly consistent stages
. First the ponds
grow quickly while the ice is impermeable. Next they drain quickly and pond
coverage shrinks as the ice transitions from impermeable to permeable. Then
the ponds grow slowly while the ice is permeable and pond water remains at
sea level. Finally, the ponds either refreeze or the floe breaks up. The
stage when ice is highly permeable is typically the longest, often longer
than the first two stages combined. This stage is particularly suitable to
model, since the ponds can be assumed to be at sea level and hydraulically
connected to the ocean. On multiyear ice, ponds also experience a growth and
a drainage stage, but often do not drain to sea level. On some occasions,
however, ponds on multiyear ice can drain to sea level as well.
In this paper we will present a simple “0-D” model for the evolution of
melt pond coverage on sea ice floes. We will assume that ice is permeable,
ponds are at sea level and hydraulically connected to the ocean, the whole
ice floe is in hydrostatic balance, and different points on the ice surface
may melt at different rates. The purpose of our model is (1) to clarify the
roles in the evolution of pond coverage played by energy fluxes, the ice
thickness, bulk ice density, ice roughness, and initial pond coverage; (2) to
provide a simple, yet accurate, way to estimate the pond coverage as a
function of time; (3) to understand the behavior of melt ponds under general
environmental conditions; and (4) to investigate different types of
qualitative behavior that can arise from differential melting and maintaining
hydrostatic balance.
also describe pond growth on permeable ice, but they
include only pond growth by lateral melt of pond walls. This contrasts with
our model, which includes pond growth by vertical changes of the
topography. Our models are different, but complementary, and we will draw
parallels between our two models when discussing the possibility of lateral
melt. Aside from , previous melt pond modeling
efforts include works by , ,
, and , who all created
comprehensive models that allowed for more realistic representations of
physical processes such as heat and salt balance, and meltwater routing and
drainage. The advantage of our model is its simplicity, which makes it
possible to clarify the roles of each of the physical parameters involved.
This paper is organized in the following way. In Sect. we
build a simple model for the evolution of pond coverage. In
Sect. , we compare the model to observations. In
Sect. we discuss realistic values of physical parameters
and solve the model numerically. In Sect. we assess the
impacts of sea ice roughness and develop a simple parameterization to
estimate mean pond coverage after a certain amount of time without solving
the model. In Sect. we analyze the model analytically to
gain a better understanding of the factors influencing pond evolution. In
Sect. we discuss lateral melt and internal melt combined
with effect of density variations. Finally in Sect. we
summarize our results and conclude. In Appendices A, B, C, and D we discuss
some of the more technical aspects of our model.
Building the simple 0-D model
In this section, we build the model for the evolution of melt pond coverage
and then solve it using realistic physical parameters. Before we proceed to
build the quantitative model, we will first state the assumptions and
discuss the physical mechanisms driving pond evolution.
Assumptions of the model
Our model focuses on the stage of pond evolution when ice is highly permeable
and all the meltwater created can be quickly removed to the ocean. The
beginning of this stage can be identified as the point in time when the
meltwater on the ice surface has drained to sea level, such that the
remaining ponds correspond to places on the ice surface that are below sea
level. We will assume that from this point on, the ponds are hydraulically
connected with the ocean, and the only way for pond coverage to increase is
for the points on the ice surface which were above sea level to sink or melt
below sea level. In reality, ponds can also grow through horizontal melting
of their sidewalls. As some observations suggest that this type of growth is
small at least on first-year ice , we
neglect it (see Sect. for further discussion).
Furthermore, we will assume that all the melt occurs at the surface or the
bottom of the ice. We thereby neglect the possibility of internal melt. We
will also assume that ice has a uniform bulk density throughout the vertical
column, and we discuss the effects of vertical nonuniformity in bulk density
together with effects of internal melt in Sect. .
Finally, we will assume that the entire ice floe is in hydrostatic balance.
The main goal of our model is to determine the fraction of the ice surface
above sea level that falls below sea level after some time. Therefore, we
focus on the vertical displacements of points on the surface of the ice in
response to melt. To this end, we define the ice topography, s(r), as
the elevation of the ice surface above sea level at the point r, and
we define melt ponds as those regions where s(r)<0. There are two
main reasons why the topography might change in response to ice melt:
First, the topography at a point r at the surface changes when
ice at that point melts (Fig. a). Here, the rate of change
of topography at a point depends only on local characteristics of that
particular point. For this reason, we will call this type of motion
“local.” Points on the surface that melt locally move “downwards,” i.e.,
to lower elevations above sea level.
Second, in order to maintain hydrostatic balance, the entire ice surface can
shift up or down in response to mass being removed above or below sea level.
Since we are assuming that the entire ice floe is in hydrostatic balance,
melting any region of ice moves the entire floe as a rigid body
(Fig. b). For this reason, we will call this type of motion
the “rigid body” motion. Melting above sea level induces an upward rigid
body motion, whereas melting below sea level induces a downward rigid body
motion. An ice floe is not a rigid body, but up to its flexural wavelength
(roughly 30 m on 1.5 m thick ice) we can approximate it as such. As the
flexural wavelength is larger than the typical scale of melt ponds (roughly
10 m), the rigid body approximation is likely good.
(a) Local displacement represents the movement of a point
on the ice surface as a result of ice melting at that particular point. It is
a function only of local ice characteristics at that point. For both local
and hydrostatic displacements the positive direction is defined as upwards.
(b) Rigid body displacement represents the motion of a floe as a
whole in an effort to maintain hydrostatic balance because melting removes
mass above or below sea level. Melting above sea level induces an
upward rigid body motion of the floe, whereas
melting below sea level induces a downward motion.
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At each point on the ice surface the change in elevation above sea level can
be calculated as the sum of these two contributions.
In our model, ponds grow in two ways, “freeboard sinking” and “enhanced
melting”:
Freeboard sinking represents the average change in freeboard height
(average height above sea level of bare ice). In this way the topography of
ice above sea level remains unchanged. Freeboard sinking should not be
confused with rigid body motion: the average freeboard height always
decreases as a response to ice thinning, whereas the rigid body motion can
point both upward and downward depending on whether mass is lost above or
below sea level. Both rigid body motion and average local melting contribute
to freeboard sinking.
Enhanced melting represents the change in the shape of the topography without
changing its average height. Ponds can grow in this way if some regions melt
faster than average. Therefore, a positive deviation in the local melt rate
can grow ponds. Conversely, a negative deviation in the local melt rate can
slow down or even reverse pond growth. Pond growth only occurs due to
topography changes near sea level. Therefore, deviations from the mean melt
rate for points high above the sea level do not influence pond evolution
since these points are correlated with points close to sea level only through
hydrostatic adjustment, which is determined by the average melt rates rather
than the deviations from the average.
Equation for the evolution of topography
We now proceed to build the quantitative model of pond evolution. Following
the above ideas, we divide the total rate of change of vertical position of
the point r on the surface of the ice,
dsdt(r), into a contribution from rigid body
motion, dsrigid bodydt, and a contribution
from local melting, dslocdt(r):
dsdt(r)=dsrigid bodydt+dslocdt(r).
Ice above sea level (asl) must hydrostatically balance ice below sea level
(bsl). We can write this hydrostatic balance as
masl=ρw-ρiρimbsl,
where masl and mbsl represent the mass of ice above
and below sea level, and ρw and ρi represent
the densities of sea water and pure ice. Throughout the paper we use
ρw=1025 kg m-3 and ρi=916 kg m-3.
The mass above and below sea level can change either because the ice melts or
because the floe moves as a rigid body, changing the proportion of ice above
and below sea level. Therefore, differentiating Eq. () and
splitting into melt and rigid body contributions, we find
dmaslmelt+dmaslrigid body=ρw-ρiρwdmbslmelt+dmbslrigid
body,
where dmasl/bslmelt/rigid body represents changes in
mass above and below sea level due to either ice melting or the entire floe
floating up or down.
The mass melted above and below sea level after some time dt is
dmaslmelt=-AbiF‾bildt,dmbslmelt=-AmpF‾mpldt-AF‾botldt,
where l=334 kJ kg-1 is the latent heat of melting,
F‾bi is the total energy flux used for melting bare ice
averaged over all bare ice, F‾mp is the total energy
flux used for melting ponded ice averaged over ponded ice, and
F‾bot is the total energy flux used for melting the ice
bottom averaged over the ice bottom. Abi, Amp, and
A are the area of bare ice, the area of melt ponds, and the area of the
entire floe, respectively.
Since floating up or down does not change the total mass of the ice, mass
changes above and below sea level due to rigid body motion are equal with an
opposite sign, dmaslrigid body=-dmbslrigid body. We can express
dmrigid body in terms of rigid body displacement of the
floe as
dmaslrigid body=ρbAbidsrigid body,dmbslrigid body=-ρbAbidsrigid body,
where ρb is the bulk ice density. This is the
density of sea ice once all the brine has drained and is always less than
ρi. We assume it to be uniform throughout the vertical ice
column, but we discuss the effects of vertical variations in ρb
in Sect. .
Substituting Eqs. (4) and (5) into Eq. (), solving for
dsrigid body, and differentiating with respect to time, we
find the rate of change of surface topography due to rigid body motion to be
dsrigid bodydt=ρiρwF‾bilρb-ρw-ρiρwAmpAbiF‾mplρb-ρw-ρiρwAAbiF‾botlρb.
The three terms in large square brackets correspond to topography change due
to bare ice melting, ponded ice melting, and ice bottom melting. Rigid body
motion depends only on spatially averaged energy fluxes, which in turn depend
on parameters such as the average insolation on the floe, the average albedo,
and the average longwave, sensible, latent, and bottom heat fluxes. If bare
and ponded ice melt only from energy absorbed by the upper surface of the
ice, the fluxes F‾bi and F‾mp
can also be written in terms of albedo as
F‾bi=(1-αbi)Fsol+Fr,F‾mp=(1-αmp)Fsol+Fr,
where αbi and αmp are the average albedos
of bare and ponded ice, Fsol is the solar flux, and
Fr is equal to the sum of net longwave, net sensible, and net
latent heat fluxes. This parameterization neglects light transmission and
assumes that all of the energy is deposited in the surface. Much of the
variation in albedo of ponded ice is due to the fact that the pond bottom is
partially transparent, and energy is deposited in the ocean instead of
directly in the ice. However, this does not make much difference in our model
since the energy deposited in the ocean is likely used for melting ice below
sea level anyway.
Local displacement, dsloc, quantifies how much the ice
surface topography changes as a result of local melt. We can determine the
local melt rate from Fsurf(r), the flux of energy used for
melting the ice surface at a point r:
dslocdt(r)=-Fsurf(r)lρb,
where the positive direction is defined as upwards. The local flux depends on
parameters such as the local albedo, the local insolation, the local
longwave, sensible and latent heat fluxes, and the angle between ice and
incoming radiation at that point.
The flux Fsurf(r) averaged over all the points on the
surface of the ice above sea level equals F‾bi:
<Fsurf(r)>=F‾bi,
where <…> represents averaging over all the points on bare ice. For
this reason, we will parameterize the rate of local melting as
dslocdt(r)=-k(r)F‾bilρb,
where k(r) is a nondimensional number that quantifies the deviation
of the melt rate at the point r from the mean melt rate of the bare
ice surface, which depends on the detailed conditions of ice and its
environment. The parameter k could be either greater than or less than 1.
Here we will take k to be constant in time, but in reality it need not be.
Finally, according to Eq. (1) we add Eqs. (6)
and () to get the equation for the evolution of the bare ice
topography. We express this in terms of melt pond fraction, x≡AmpA:
dsdt(r)=-(k(r)-1)F‾bilρb-ρw-ρiρw1lρbF‾bi+x1-xF‾mp+11-xF‾bot.
Here, we split the equation into two terms, enclosed by the square brackets.
The first term represents the local deviation from the average surface melt
rate, which changes the general shape of the topography while preserving its
average height above sea level. We identify this term with enhanced melting.
The second term represents a global shift of the average elevation above sea
level due to freeboard sinking.
In this way, the topographic evolution equation can be split into two terms,
enhanced melting and freeboard sinking:
dsdt=dsemdt+dsfsdt,
where dsemdt and
dsfsdt are contributions from enhanced
melting and freeboard sinking, and they correspond to the first and second term
of Eq. ().
Model for the evolution of pond coverage
We now need to relate the vertical displacements near the sea level to the
change in area of the melt ponds. To this end we define the hypsographic
curve, s(xh), which relates the elevation above sea level, s, to the
percent of ice surface below that elevation, xh (Fig. ).
Such curves have been measured and reported on several occasions (e.g., Fig. 8
of , or Fig. 8 of ). If the ice is highly
permeable, the melt pond fraction, x, can be inferred from a hypsographic
curve as the intersection of sea level with the curve. Since ponds are
hydraulically connected with the ocean, the average freeboard height of bare
ice, h, depends on the pond fraction. The average freeboard height, h,
can be expressed in terms of the ice thickness H and the pond fraction as
h=ρw-ρiρwH1-x.
Hypsographic curves showing the percentage of the sea ice surface
that is lower than a particular elevation. Pond coverage on highly permeable
sea ice can be inferred from here as the intersection of sea level
(horizontal blue line) with the hypsographic curve. (a) A
hypsographic curve measured by on 25 June 2011 (solid black
line) and a hypsographic curve measured during SHEBA along a 100 m long
“topography profile 1” on 10 July 1998 (black dashed line). The vertical
dashed lines represent the pond coverage, assuming that ice is permeable. The
red line represents a fit to the part of the hypsographic curve above sea
level with a tangent function, Eq. (A1). (b) Adjusted hypsographic
curves for different initial pond coverage and the same ice thickness.
(c) Adjusted hypsographic curves for the same initial pond coverage
and different ice thickness. (d) Hypsographic curves for different
shape parameters, p1 and p2, defined and discussed in Appendix A,
Eq. (A1). Parameter p1 controls the amount of curvature, while p2
controls the position of the inflection point of the tangent function.
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Here, the average freeboard height is defined as the elevation of the ice
surface above sea level averaged over bare ice. For two ice floes of the same
thickness, the one with higher pond coverage will also need to have a higher
average freeboard in order to maintain hydrostatic balance.
The above sea level part of every measured hypsographic curve we tested can
be fit relatively well with a tangent function (Fig. a, red
line). We will assume that this fit holds for a wide range of different sea
ice floes and use it to initialize our model with different physical
parameters. We give the exact form of this function in Appendix A (Eq. A1).
To get a hypsographic curve for a particular initial pond fraction, xi,
and ice thickness, H, we set it to zero at the initial pond coverage,
s(xh=xi)=0, and rescale it vertically to get a freeboard that
hydrostatically balances the floe. The topography below sea level is not
important for the evolution of pond coverage if the pond coverage grows, and
we replace it with a straight line.
We show several curves for different initial ice thickness and initial pond
coverage in Fig. b and c. We note that the initial pond
fraction, xi, corresponds to the pond fraction when ice first becomes
permeable. Once we choose xi and H, the tangent function Eq. (A1) has
only two unconstrained parameters, p1 and p2, that determine the exact
shape of the curve. Knowing additional physical parameters, such as ice
roughness, we can constrain additional parameters of this curve. Throughout
this paper we will mostly use p1 and p2 that fit the measurements of
the hypsographic curve made by for 25 June 2011 or the
measurements made during the SHEBA mission along the topography profile “1”
on 10 July 1998. However, when examining the effects of sea ice roughness, we
will vary these parameters to get curves of different shape. Several examples
of hypsographic with different p1 and p2 are shown in
Fig. d.
In the case of pure freeboard sinking the overall shape of the hypsographic
curve does not change as the ice melts. Instead the whole curve is shifted
following a displacement of dsfs
(Fig. a). We can calculate the resulting change in pond
coverage as
dxdt=dxhds(x)dsfsdt,
where dsfs is the vertical displacement of the bare ice
topography due to freeboard sinking (as determined by the second term in
Eq. ), and dxhds(x) is the change
in pond fraction for a vertical shift of the ice surface of
dsfs when the pond fraction is equal to x. It is equal
to the reciprocal of the derivative of the hypsographic curve, s(xh),
evaluated at xh=x. Substituting
dsfsdt from Eq. () we find
dxdt=dx^hds^(x)Sbi+Smpx^1-x^+Sbot11-x^,
where x^≡xxi and 1-x^≡1-x1-xi are the pond and bare ice fractions normalized by the
initial pond and bare ice fractions, and
dx^hds^(x)≡h1-xidxhds(x) is the nondimensional slope of the
hypsographic curve. We have defined the strengths of pond growth by
freeboard sinking due to melting bare, ponded, and ice bottom,
Sbi, Smp, and Sbot, as
Sbi≡(1-xi)2F‾biHlρb,Smp≡(1-xi)xiF‾mpHlρb,Sbot≡(1-xi)F‾botHlρb.
Explanation of different models of pond growth. Models evolve a
hypsographic curve, s(xh), above sea level to find the pond coverage
evolution. Evolution of the hypsographic curve below sea level is not
relevant for pond growth and, apart from the 1-D model, is not captured well
in these models. (a) Freeboard sinking shifts the entire
hypsographic curve downward following a displacement of
dsfs. (b) Enhanced melting acts on a constant ice
fraction, δ, and there is no freeboard sinking. The hypsographic curve
changes only between xh=x and xh=x+δ and remains unchanged
otherwise. After a time Δt=s(x+δ)dsem/dt pond coverage grows by
δ. The 0-D model, Eq. (), assumes that the total pond
evolution is the sum of pond evolution due to such enhanced melting and
freeboard sinking (panel a). (c) The 1-D model prescribes a
melt rate at each point on the hypsographic curve as a function of height
above sea level, dsdt(s). (d) A simplified
model that assumes both freeboard sinking and enhanced melting (Appendix B).
Enhanced melting occurs only below height Δs. After some time, the
fraction of ice affected by enhanced melting, δ, becomes constant,
meaning that a constant fraction model (panel b) and a constant
height model are equivalent if δ and Δs are related
appropriately.
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The nondimensional factors x^, 1-x^, and
dx^hds^(x) are chosen to be of the order
unity, so that Sbi, Smp, and Sbot control
the strengths of pond growth by melting bare ice, melting ponded ice, and
melting ice bottom. The reciprocals of the strengths represent the timescales
of the growth modes.
The set of parameters needed to describe pure freeboard sinking can be
further reduced by rewriting Eq. () as
dxdt=dx^hds^(x)S1x^1-x^+S211-x^,
where S1≡Smp-xiSbi/(1-xi) and S2≡Sbot+Sbi/(1-xi) represent a minimal set of
parameters needed to describe pure freeboard sinking. However, these
parameters do not have a clear physical interpretation, and we will
henceforth focus only on Sbi, Smp, and
Sbot.
Next we need to consider the contribution from enhanced melting. Before doing
so we need to make some assumptions about the nature of enhanced melt. There
are multiple physical processes that can cause the melt rate to deviate from
the mean. One process that stands out as being particularly important is
albedo decrease due to ice wetting: ice close to sea level will likely be wet
and therefore have a lower albedo compared to ice higher up. The deviation
from the mean melt rate in this case depends primarily on the height above
sea level. Another potential contribution to height-dependent enhanced melt
may effectively come from random fluctuations in the melt rate around the
average: ice near the sea level has a higher probability of falling below sea
level due to random fluctuations than ice higher up. After falling below sea
level, ice becomes ponded, melts faster, and is unable to return to its
previous position. Other processes, such as lateral melt, may not depend on
height above sea level, but for now we neglect this possibility (see
Sect. for discussion).
Because of the processes described above, we will assume that the deviation
from the mean melt rate, k(r)-1, depends only on height above sea
level, s. In this scenario, we need to consider enhanced melting together
with freeboard sinking, as freeboard sinking constantly supplies new ice to
low elevations to be affected by enhanced melting. Effects of enhanced
melting and freeboard sinking can be approximately separated if, instead of
height dependence, enhanced melting is constrained to act on a fixed fraction
of bare ice. In this case, a constant fraction of bare ice that would
experience enhanced melting would evolve, at least approximately,
independently of freeboard sinking.
Therefore, we will consider two cases of enhanced melting. Firstly, we will
consider a height-dependent enhanced melting. In particular, we will assume
that k(0<s<Δs)≡k and k(s>Δs)≡1, where Δs is a height above which there is no enhanced melting and below which
enhanced melting is constant k>1. This is the case we ultimately wish to
describe. We describe a potential model for pond growth under this assumption
in Appendix B and Fig. d. However, from a practical
viewpoint, it is simpler to consider enhanced melting which acts upon a fixed
fraction of bare ice. In this case, we will assume that k(x<xh<x+δ)≡k and k(xh>x+δ)≡1, where δ is a
fraction of ice affected by enhanced melting (Fig. b).
In Appendix B, we show that, if δ is appropriately chosen, a
height-dependent model and a fixed fraction model become equivalent.
Therefore, we will first solve a model assuming a fixed δ and no
freeboard sinking and then relate it to a fixed Δs model by choosing
the appropriate δ.
We note that the assumption that k(r)=1 high above the sea level
and k(r)>1 near the sea level is strictly not true since averaged
over all of bare ice k(r) needs to equal 1. However, it is
approximately true if Δs or δ are small, such that the area
where k(r)≠1 is small compared to the total area of bare ice.
Also, we have assumed k(r)=1 high above the sea level without loss
of generality, since deviations from the mean melt rate high above the sea
level are not important, as only ice close to sea level may become ponded.
Now we proceed to consider the case of “pure enhanced melting” that assumes
a fixed fraction of the ice, δ, melts, and there is no freeboard
sinking (Fig. b). If there is no topographic variation
above sea level, and the entire ice floe above sea level has the same height,
h, the pond coverage would grow by δ after a time Δt=hdsem/dt, where
dsem/dt is the rate of change of topography due to
enhanced melting as determined by the first term of Eq. ().
Therefore, the pond growth rate in this case would be ΔxΔt=δhdsemdt. If there is
non-negligible topography above sea level described by the hypsographic
curve, the time Δt it takes for pond coverage to grow by δ
would be Δt=s(xh=x+δ)dsem/dt. Here, s(xh=x+δ) is
the original hypsographic curve evaluated at xh=x+δ. We will
assume this expression generally holds for enhanced melting. Thus, we arrive
at the expression for pond growth due to pure enhanced melting with fixed
δ:
dxdt=δs(x+δ)dsemdt.
If δ is small compared to the variation in the hypsographic curve, we
can substitute s(x+δ) with s(x). This is only not justified near
the beginning of the melt, when s(x)≈0. Substituting
dsemdt from Eq. () we find
dxdt=Sem1s^(x+δ),
where s^(x)≡s(x)h is the nondimensional hypsographic
curve, and the strength of the enhanced melting, Sem, is defined
as
Sem≡ρwρw-ρi(1-xi)δ(k-1)F‾biHlρb.
Ultimately, however, our goal was to describe the height-dependent enhanced
melting. In Appendix B, we showed that such a model can be approximated with
a fixed fraction model, if we appropriately relate δ and Δs.
Here we simply state the result:
δ=ρwρw-ρi2Δs(1-xi)23H1+dsemdsfs.
Here, dsemdsfs represents the
ratio of the topographic rate of change due to enhanced melting to freeboard
sinking and is given by
dsemdsfs=ρwρw-ρi|F‾bi|(k-1)|F‾bi|+xi1-xi|F‾mp|+11-xi|F‾bot|,
where |F‾| are the
representative values of energy fluxes, e.g., their time averages. Therefore,
the strength of height-dependent enhanced melting becomes
Sem=ρwρw-ρi22Δs(1-xi)3(k-1)F‾bi3H2lρb1+dsemdsfs.
We have made a number of assumptions in deriving the expression for enhanced
melting. Below we compare this model to a more complicated “1-D” model and
show that all these assumptions are justified. We also show that if the
function describing the local melt rate, k(s), has a nontrivial dependence
on height above sea level, parameter Sem is better replaced with
a parameter:
<Sem>≡ρwρw-ρi22(1-xi)3F‾bi3H2lρb∫0∞k(s)-11+dsemdsfs(s)ds.
(a) A comparison between pond evolution in the 0-D model
and the 1-D model. The black curve represents the 0-D model. The blue, green,
and red curves represent the 1-D model for different functions k(s) shown
in panel (b). These different functions were chosen such that the
integral parameter <Sem> (Eq. ) is the same as for
the 0-D model. The yellow curve represents the 1-D model where enhanced
melting acts on a constant fraction of bare ice, δ, chosen according
to Eq. (). The magenta curve represents the 1-D model with pond
albedo varying with depth. There is significant agreement between all of the
curves, suggesting that the simplifications made in the simple model were
justified. Since including variable pond albedo does not change the pond
evolution significantly, this detail can be neglected when estimating the
pond coverage on permeable ice. (b) The blue, green, and red lines
represent functions k(s)-1 used to run the 1-D model.
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In this way, we have separated the effects of freeboard sinking and enhanced
melting. Finally, we will assume that contributions from freeboard sinking
and enhanced melting can be added independently. Therefore, we solve
Eq. () for pure freeboard sinking and Eq. () for enhanced
melting independently, and we add them together to get the full evolution of
pond coverage, x(t):
x(t)=xfs(t)+xem(t)-xi,
where xfs(t) and xem(t) are solutions to
Eqs. () and (), both forced using the same parameters and
initialized with the same initial pond fraction xi. This concludes the 0-D
model.
Equation () represents a sum of solutions to two simple ordinary
differential equations, in which the rate of change of pond fraction depends
on the pond fraction. Here, we have reduced the number of parameters from the
original 10 (H, xi, ρb, F‾bot,
Fsol, Fr, αbi, αmp,
k, and Δs) to 4 (Sbi, Smp,
Sbot, and Sem). The strengths of freeboard sinking,
Sbi, Smp, and Sbot, depend only on the
parameters that are available in GCM simulations and are relatively easily
measured in observational studies. The enhanced melting strength,
Sem, however, also depends on the difficult-to-measure parameters
k and Δs that describe the melt rate near the sea level and may
also have contributions from processes that are not height dependent.
Furthermore, as we discuss below, ice roughness can also play an important
role in pond evolution. With reliable constraints on these parameters, our
model would be a useful parameterization in GCMs for pond growth after ice
becomes permeable.
Testing the model
In order to test the assumptions we made to simplify the model, we have
developed a “1-D” model in which we explicitly determine pond evolution
when both freeboard sinking and enhanced melting are happening
simultaneously. Apart from resolving the melt rates in one dimension, the
underlying assumptions for the 1-D model are essentially the same as for the
simple model. For this reason, we simply give an outline for this model,
without discussing it in much detail.
In the 1-D model, we evolve the hypsographic curve by prescribing a melt
rate, dsloc, to each point on the hypsographic curve
depending on the height above sea level (Fig. c). The
hypsographic curve high above sea level melts at a uniform rate, whereas the
hypsographic curve slightly above sea level melts at an enhanced rate. Parts
of the curve below sea level melt at a uniform rate determined by the flux
used for melting ponded ice, F‾mp. Finally, hydrostatic
adjustment is calculated by finding the ice thickness directly at each time
step and placing the floe in hydrostatic balance. The evolution of pond
coverage obtained from this model is shown in Fig. a.
The comparison with the simple 0-D model is excellent.
The 1-D model allows us some freedom to test the detailed assumptions of the
0-D model. First, we can test how the functional form of k(s) affects the
pond evolution (Fig. b). The functions k(s) were
chosen such that they all have the same integral parameter <Sem>
defined in Eq. (). Figure a shows that
in each of these cases the evolution of pond coverage proceeds nearly
identically. Second, we can test the difference between an assumption that
enhanced melting acts below a constant height Δs and an assumption
that enhanced melting acts on a constant fraction of ice, δ. The
yellow line in Fig. a shows that if δ and
Δs are chosen according to Eq. (), both assumptions yield
very similar results. Finally, we can test the effects of varying pond
albedo. In reality pond albedo decreases as the ponds deepen. We assume a
dependence of pond albedo on pond depth reported in Table VII of
for mean broadband albedo. The magenta line in
Fig. a shows that allowing for pond albedo to vary has
a negligible effect on pond evolution.
We should note that, when both freeboard sinking and enhanced melting occur
simultaneously, the agreement between the 0-D model and the 1-D model becomes
poor if the hypsographic curve is convex (e.g., Fig. d, blue
curve), and the 0-D model should be used with care. Happily, the measured
hypsographic curves are mostly concave, in which case the agreement between
the two models is excellent.
Numerical solutions
We now solve Eq. () numerically to gain intuition about the
behavior of our model. We use a set of realistic parameters we will
henceforth refer to as the “default parameters.”
For shortwave, longwave, latent, and sensible heat fluxes, we use values
inferred by using hourly measurements from the SHEBA
mission. We use the bottom heat flux inferred from measurements of ice bottom
ablation during the SHEBA mission . The albedo of bare
ice can vary between 0.5 and 0.7 , while the albedo of
melt ponds can vary between 0.1 and 0.6, depending on pond depth and
conditions of ice at the pond bottom
. Here we prescribe a
default bare ice albedo of 0.55 and a default pond albedo of 0.2. We use
a realistic bulk ice density of ρb=850 kg m-3
. We use an initial ice thickness of 1.5 m and use the
first-year ice topography measured by adjusted for the
prescribed ice thickness and initial pond fraction (usually xi=0.2). We
will assume enhanced melting is entirely due the albedo dependence on height
above sea level. Some preliminary results based on field measurements of bare
ice albedo on first-year ice suggest that albedo changes from around 0.3
near sea level to around 0.55 at a height of around 10 cm above sea level,
after which the correlation between albedo and surface elevation tapers off
(Chris Polashenski, personal communication, 2017). Using such an albedo and
the average values of shortwave, longwave, latent, and sensible heat fluxes,
we can estimate the rate of melt as a function of height above sea level,
k(s)=F‾(s)F‾bi. Using
Eq. (), we can then find the integral parameter
<Sem>. We choose Δs=6 cm and k=1.7 to correspond
to the same integral parameter. We should note that there is significant
scatter in the data, and measurements correspond to only one study.
Therefore, this is a rough estimate of enhanced melting, but it is likely of
the correct order of magnitude.
A comparison between measurements of pond fraction made during SHEBA
along the albedo line (red line), along a topography profile (blue dots), and
our model (black line). The blue dots have been shifted downward by 0.05 to
make a more obvious comparison between albedo line and topography profile
trends. The black dashed line is the contribution to our model from freeboard
sinking and the black dotted line is the contribution from enhanced melting.
Ponds grow almost entirely due to enhanced melting as a result of the steep
topography of multiyear ice.
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Figure a shows the solution to Eq. () for
different initial conditions. We can see that ponds grow more rapidly when
the initial pond coverage is lower, and the pond evolution curves cluster
together as time progresses. This is because lower initial pond coverage
corresponds to lower initial freeboard height, making the pond growth more
rapid. The dashed line corresponds to the solution using the fluxes
time-averaged over the 30-day run. The solutions using the averaged fluxes
are very similar to the ones using time-varying fluxes, meaning that daily
and even monthly variations in the forcing have little effect on pond growth.
This insensitivity to short timescale variations in the forcing means that
pond coverage evolution may be faithfully represented in the large-scale
models, as it would not be affected by the coarse timescales of those
models. Henceforth, we will use the time-averaged fluxes.
A larger ice thickness means a higher freeboard. For this reason, ponds grow
more slowly on thicker ice. Because the pond growth rate is inversely
proportional to ice thickness, pond coverage is more sensitive to variations
in ice thickness when the ice is thin (Fig. b). In
Fig. b we see that a 0.5 m difference in the initial
ice thickness (between a floe 1.5 m and a floe 2 m thick) can mean a
20 % difference in pond coverage at the end of the melt season.
Numerical solutions to Eq. () with parameters varied
around the defaults described in the text. (a) Varying initial pond
coverage. Solid lines represent solutions using full time-varying fluxes,
while dashed lines represent solutions using time-averaged fluxes. The two
solutions are very similar, so we subsequently use only the time-averaged
fluxes. (b) Varying ice thickness. Ponds grow slower on thicker
floes. (c) Varying pond and bare ice albedo. Different colors
represent different bare ice albedos, and full, dotted, and dashed lines
represent different pond albedos. A change in bare ice albedo has a much
larger effect on pond fraction than the same change in pond albedo.
(d) Varying the Δs and k. For k=0.8, the ponds shrink.
However, pond evolution for k<1 is not represented well in our model, so
this curve serves only as an illustration.
![]()
Figure c shows the dependence of pond coverage on
albedo. A variation of 0.1 in bare ice albedo has a much larger effect on
pond evolution than the same change in pond albedo. The reason is that
melting ponded ice only affects pond coverage through downward rigid body
motion of the floe, whereas melting bare ice grows the ponds through both
enhanced melting and freeboard sinking. Furthermore, when pond coverage is
low, rigid body motion due to ponded ice melting is less efficient than that
due to bare ice melting because it is proportional to melt pond fraction.
Exploring the effects of sea ice roughness. (a) Pond
evolution due to pure freeboard sinking for hypsographic curves with
different shape parameters p1 and p2. The x axis shows nondimensional
time t^=t(Sbi+Smp+Sbot)1-xi. Color represents normalized roughness,
σ^, with blue colors corresponding to small σ^ and
red colors corresponding to large σ^. Thick red solid line
represents pond evolution on the measured first-year ice hypsographic curve,
and the thick red dashed line represents pond evolution on the measured
multiyear ice hypsographic curve. All else equal, rougher ice has a larger
pond fraction. (b) Pond evolution due to pure enhanced melting for
hypsographic curves with different shapes. The x axis shows nondimensional
time t^=tSem1-xi. Cartoon examples of
hypsographic curves and their approximate positions along the σ^
axis are also shown.
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The parameters controlling the strength of enhanced melting are the least
constrained parameters in our model. In Fig. d we show
the dependence of pond evolution on the height below which enhanced melting
is active, Δs. Exploring a range of realistic values for Δs,
0<Δs<15 cm, we find that the pond fraction at the end of the
melt season can vary by about 30 %. This difference would be larger if we
chose a smaller ice thickness. The effects of changing k are relatively
small, so long as k is large enough (not shown). For example, using current
parameters, pond coverage evolution becomes fairly insensitive to k when k>1.5. Smaller values of k, however, can significantly impact pond
evolution. If k is sufficiently smaller than 1, Sem can become
negative and the pond coverage can stop growing. In this case, ice near the
sea level melts slowly enough such that an upward rigid body movement due to
melting ice high above sea level pushes the ice near sea level upwards,
preventing pond coverage growth. The evolution of such a pond coverage cannot
be represented well in our model since the equation for enhanced melting
becomes invalid in this case, and the blue curve in
Fig. d serves therefore simply as an illustration.
Pond evolution is slower on smoother ice
The evolution of pond coverage in our model depends on the detailed shape of
the hypsographic curve, which is not captured by the strengths of freeboard
sinking and enhanced melting. As we show below, pond coverage is sensitive to
such details and in particular to ice roughness. Below we will introduce the
“effective strengths”, S*, which approximately capture the effects of
roughness and allow us to estimate mean pond coverage after a period of time.
Using effective strengths, we will demonstrate how multiyear ice topography
suppresses pond growth by freeboard sinking, while first-year ice topography
permits it.
In the tangent function parameterization, Eq. (A1), the exact shape of the
hypsographic curve is determined by parameters p1 and p2. Here, we will
not discuss these parameters individually but will rather focus on often
measured bare ice roughness, σ, defined as the standard deviation of
surface elevation of ice above sea level:
σ≡∫xi1s2(xh)dxh1-xi-h212.
We will use the nondimensional form of bare
ice roughness, defined as σ^≡σh. Typically,
a concave hypsographic curve (e.g., Fig. d, red curve) will
have a small σ^, whereas a convex hypsographic curve (e.g.,
Fig. d, blue curve) will have a high σ^.
During the permeable stage, all else equal, ponds will grow more rapidly on
rougher ice, since a larger fraction of ice is close to sea level. This is
not true on impermeable ice, as meltwater filling deep topographic lows on
rough ice will cover a smaller area relative to the same amount of meltwater
filling shallow topographic lows on smooth ice. For this reason, the initial
pond coverage will likely be smaller on rougher ice due to a smaller pond
coverage during the impermeable stage.
Figure shows the pond coverage evolution due to pure
freeboard sinking (Fig. a) and pure enhanced melting
(Fig. b) for hypsographic curves with different parameters
p1 and p2 and all other parameters kept constant. For each choice of
p1 and p2, we find the normalized bare ice roughness, σ^,
represented by the color of the curves. Blue colors correspond to low
roughness and red colors to high roughness. Pond evolution on measured
topographies (Fig. a) is also shown. We can see that although
roughness does not fully determine the pond evolution, it is a viable proxy
for how pond coverage will evolve, with high roughness curves typically
having a higher average pond coverage.
We wish to quantify the effect of roughness by its impact on the mean pond
coverage. In particular, we hope to find the “effective strengths”,
S*(σ^), which include the roughness effects and allow us to
easily estimate the average pond coverage after some time t:
<x(t)>≈12S*t+xi,
where <x(t)>≡∫0tx(t)dtt. Effective strengths
are proportional to strengths of freeboard sinking and enhanced melting we
derived in Sect. . In general they themselves may depend on
time and are independent of time only if pond coverage evolution is linear,
x(t)=St+xi, in which case S*=S, where S is either Sfs≡(Sbi+Smp+Sbot) in the case of
freeboard sinking or Sem in case of enhanced melting.
In Appendix C, we describe the procedure to estimate the effective strengths
as functions of nondimensional roughness and time. Here, we only state the
result:
Sfs*≈1.3σ^2Sbi+Smp+Sbot,Sem*≈1+2t^em-32σ^Sem,
where Sfs* is the effective strength of freeboard sinking,
Sem* is the effective strength of enhanced melting, and
t^em≡Semt1-xi is the
nondimensional time of pond evolution due to enhanced melting. The terms in
square brackets represent the corrections due to roughness. If both freeboard
sinking and enhanced melting occur simultaneously the total effective
strength is the sum of these two, S*=Sfs*+Sem*.
Knowing the effective strengths allows us estimate the mean pond coverage
after a period of time without having to run the model.
Roughness has a different effect on freeboard sinking and enhanced melting.
Freeboard sinking is roughly independent of time and proportional to the
square of nondimensional roughness. Therefore, it is very sensitive to
variations in roughness: doubling the ice roughness roughly quadruples the
mean pond coverage due to freeboard sinking after some time. Enhanced melting
depends roughly linearly on roughness. However, as roughness tends to zero,
the effective strength remains nonzero, Sem*(σ^→0)→Sem. Therefore, ponds on smooth ice
grow primarily due to enhanced melting. Effective strength also depends on
the nondimensional time, t^, and is higher and more sensitive to
variations in roughness early in the melt season.
Multiyear ice topography shown in Fig. a, dashed line, has
σ^≈0.25 and is significantly smoother than first-year ice
topography shown in Fig. a, solid line, which has
σ^≈0.55. From Eq. (29) it follows that freeboard sinking
on multiyear ice is roughly 5 times less efficient in growing the ponds than
on first-year ice.
Analyzing the 0-D model yields useful insight into factors influencing the pond evolution
Extracting the dependence of a desired property on physical parameters and
understanding its scaling is the main strength of our model. These types of
relationships would be difficult to obtain in a more complex model.
The parameters Sbi*, Smp*, Sbot*, and
Sem* control the mean rates of pond growth by melting different
regions of ice. Roughly, they represent the amount of pond growth per unit
time by freeboard sinking due to melting bare ice, freeboard sinking due to
melting ponded ice, freeboard sinking due to melting ice bottom, and enhanced
melting. Knowing these parameters allows us to estimate mean pond coverage
after a period of time with significant accuracy without having to run the
numerical model. Moreover, analyzing them can yield useful insight into the
behavior of melt ponds under general circumstances.
We can estimate the change in magnitude of the strength of each of the growth
modes when a physical parameters p changes by Δp as
ΔSi*=∂Si*∂pΔp,
where ΔSi* is the change in magnitude of the effective strength of
the ith growth mode. This equation holds so long as the change in the
physical parameter is not too large. A change in pond growth rate can then be
estimated as ΔS*=∑iΔSi*. Then, using
Eq. (), we can roughly estimate a change in mean pond fraction,
Δ<x>, after some time, Δt, following a change in physical
parameter, p, as Δ<x>≈12ΔS*Δt. This
provides a means to estimate changes in mean pond coverage under different
environmental conditions.
Ponds are more sensitive to changes in bare ice albedo than changes in pond albedo
We will illustrate the use of effective strengths using an example where we
vary the ice and pond albedos. If the bare ice albedo changes by Δαbi, the change in growth rate would be roughly
ΔS*=Sbi*+ρw-ρiρwdsem/dsfs2+(k-1)1+dsemdsfs(k-1)Sem*×FsolF‾biΔαbi≈-0.91monthΔαbi.
If the melt pond albedo
changes by Δαmp, the change in growth rate would be
roughly
ΔS*=-Smp*+ρw-ρixidsem/dsfs2F‾mpρw1+dsemdsfs(k-1)(1-xi)F‾biSem*×FsolF‾mpΔαmp≈-0.21monthΔαmp.
It follows from these estimates that after a month the mean pond fraction
would differ by roughly 4.5 % for a bare ice albedo difference of 0.1
and by around 1 % for a pond albedo difference of 0.1. Therefore,
variation in pond albedo affects pond evolution roughly 5 times less than
variation in bare ice albedo. This explains our observation from
Fig. c that pond evolution is much more sensitive to
variations in bare ice albedo than to variations in pond albedo. In this way,
we also extract the dependence of sensitivity on physical parameters. A major
difference between the two sensitivities is their dependence on the initial
pond coverage: the sensitivity to pond albedo is proportional to xi,
whereas the sensitivity to bare ice albedo is proportional to 1-xi. In the
above example we used xi=0.2, which explains most of the large
difference between the two sensitivities. If the pond coverage were higher,
variations in the pond albedo could become more important than variations in
bare ice albedo. For example, assuming no enhanced melting, the sensitivity
to pond albedo would become greater than the sensitivity to bare ice albedo
at 50 % pond coverage ΔSmp*ΔSbi*=xi1-xiΔαmpΔαbi.
(a) Dependence of growth rate on pond coverage for
different modes of pond growth. The y axis shows the growth rate,
dxdt, for each of the growth modes calculated using
the default parameters and xi=0. Pond growth rate for bare ice melting
(blue line) first increases up to a certain pond coverage and then decreases.
Ponded ice melting (green line) increases with pond coverage from
dxdt=0 at x=0 to very high values at high pond
coverage. The ice bottom melting rate (red line) gradually increases with
pond coverage. The vertical enhanced melting rate (cyan line) decreases with
pond coverage. The black line represents a realistic combination of the four
growth modes and shows that pond growth is dominated by enhanced melting
early in the season and by freeboard sinking late in the season. The dashed
magenta line represents lateral melting estimated using parameters described
in Sect. . (b) Solutions to Eq. ()
when only one of the growth modes is active. The x axis shows the
normalized time, where 0 corresponds to the beginning of the melt and 1 to
entire floe being flooded.
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Under global warming, pond feedback could lead to significant ice thinning
We now use the effective strengths to roughly estimate the impact of global
warming on the pond coverage. At high latitudes, feedbacks due to changes in
albedo, the atmospheric lapse rate, and clouds can amplify the forcing due to
global warming . For this reason forcing at high
latitudes is generally larger than direct radiative forcing due to an
increase in CO2 concentration. In a global warming scenario, the pond
growth rate would increase because the ice melts faster but also because ice
at the beginning of the melt would be thinner. We can emulate a global
warming scenario by increasing the flux Fr by a certain amount, ΔFr, and by assuming that the initial ice thickness decreases by ΔH≡∂H∂FrΔFr, where ∂H∂Fr is the ice thinning per 1 W m-2 of warming.
Therefore, we split the change in pond growth rate due to global warming,
ΔS*, into a contribution from direct forcing, ΔSF*, and
a contribution from ice thinning, ΔSH*. Using the above formalism, we find
ΔSF*≡∑i∂Si*∂FrΔFr=Sbi*F‾bi+Smp*F‾mp+ρw-ρiρw(dsem/dsfs)2+(k-1)(1-xi)(1+dsemdsfs)(k-1)(1-xi)Sem*F‾biΔFr≈0.5%Wm-2×monthΔFr,ΔSH*≡∑i∂Si*∂H∂H∂FrΔFr=-Sbi*+Smp*+Sbot*+2Sem*1H∂H∂FrΔFr≈1.9%W/m2×monthΔFr,ΔS*≡ΔSF*+ΔSH*≈2.4%Wm-2×monthΔFr.
The numbers in Eq. (33) were obtained using the default values of the
parameters, and ∂H∂Fr=-0.05 m3 W-1
roughly estimated using the model. This means that after
a month's growth global warming would increase mean pond coverage by roughly
1.2 % per 1 W m-2 of warming. Nearly half of this increase in the
mean pond coverage comes from an increase in the strength of enhanced melting
due to ice thinning. Simulating a 30-day melt numerically using our model
predicts an increase in mean pond coverage with forcing at a rate of
1.5 % per 1 W m-2 of warming for small forcing (ΔFr≈0), which confirms the approximate validity of our linearization.
For larger forcing, the sensitivity of pond coverage to forcing increases
because the ice thins. Our linearized estimate, Eq. (33), also gives the
dependence of the sensitivity on physical parameters. In a likely scenario
where the forcing is around 10 W m-2, our estimate predicts that after
a month mean pond coverage would increase by around 15 %, which
corresponds to around 12 cm of ice thinning solely due to the pond feedback.
Ice thinning after a month directly due to forcing would be only around
9 cm, meaning that the pond feedback must be taken into account to
understand ice thinning under global warming. Increased forcing could also
lead to changes in initial pond coverage, changes in ice roughness, or
changes in Δs or k. We ignored these feedbacks, as we have no way
of reliably estimating ∂p∂Fr for these
parameters.
Different growth modes yield different pond evolution
Each of the four growth modes has different effects on the pond coverage. We
will now look in detail at each of the growth modes, their effect on the pond
evolution, and their scaling with physical parameters. Figure
shows the dependence of growth rate on pond fraction and solutions to
Eq. () when only one of the strengths is nonzero, assuming a
first-year ice topography. Figure shows the evolution of
pond coverage distribution when only one of the strengths is nonzero.
All modes of growth depend in the same way on the bulk ice density,
ρb. Each of the strengths is inversely proportional to
ρb, meaning that ponds grow faster on ice with a lower bulk
density. The effect is, however, modest: within a reasonable range of
916 kg m-3 >ρb>750 kg m-3, pond growth rate
can vary by at most 20 %.
In this figure we have evolved an ensemble of 105 floes with
varying initial pond coverage according to Eq. () when only one of
the growth modes is active. Red curves represent the initial pond fraction
distribution, blue curves represent the pond fraction distribution after a
time, t, while the green curves represent the pond fraction distribution
after 2t. A time used in panel (a) is t=121-xiSbi, in panel (b) it is t=161-xiSmp, and in panels (c) through (f) it is t=141-xiS, where xi is the mean pond fraction of the
initial distribution and S is an appropriate strength. We show how
different growth modes have different effects on the pond fraction
distribution. (a) Bare ice melting first narrows the distribution
and then widens it. (b) Ponded ice melting widens the distribution.
(c) Bottom ice melting narrows the distribution, while the mean of
the distribution increases at an increasing rate. (d) Enhanced
melting narrows the distribution, while the mean of the distribution
increases at a decreasing rate. (e) Using realistic parameters, the
pond distribution slowly narrows and accelerates. (f) Due to lateral
melting, pond coverage distribution does not change width, and the growth is
linear.
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We will first discuss freeboard sinking. Common to all modes of freeboard
sinking is the dependence on ice thickness. Each freeboard sinking growth
mode is inversely proportional to the ice thickness, Sfs*∝1H, meaning that, all else equal, ponds grow proportionally slower
on thicker ice.
Although ice roughness may have a different effect on each of the individual
modes of freeboard sinking, for simplicity we will assume that they are all
affected by roughness in the same way, as parameterized in Eq. (29). In that
case, each of these strengths is roughly proportional to the square of the
nondimensional ice roughness, Sfs*∝σ^2,
meaning that pond growth due to freeboard sinking is suppressed on smooth
ice.
We will now focus on individual components of freeboard sinking. The
parameter Sbi* controls pond growth by freeboard sinking due to
melting bare ice. On first-year ice, owing to the shape of the hypsographic
curve, the pond growth rate by bare ice melting increases up to a certain
pond coverage and decreases afterwards (Fig. , blue line).
Sbi* is proportional to the flux F‾bi and
depends on the initial pond coverage as Sbi*∝(1-xi)2. The quadratic dependence on initial bare ice fraction means that
ponds on floes with less initial pond coverage grow faster. It also means
that floes that start off less ponded can at some point become more ponded
than floes that start off more heavily ponded. We can see this in
Fig. a, where the pond coverage distribution narrows up to
a certain point, after which it starts to widen again because floes with
lower xi overtake the floes with higher xi. Using the default values of
physical parameters of F‾bi=85 W m-2, H=1.5 m, xi=0.2, and σ^=0.55, we get Sbi*≈0.13 month-1.
The parameter Smp* controls pond growth by freeboard sinking due
to melting ponded ice. The pond growth rate increases with pond fraction from
0 at x=0 to very high values at high pond coverage and can be the
dominant mode of pond growth if the pond coverage is high enough
(Fig. , green line). For this reason, giving a representative
number to pond growth rate, such as Smp, is only meaningful if
the melt season is short enough such that pond coverage during that period
does not change substantially. The dependence on initial pond coverage is
Smp*∝xi(1-xi). For this reason the pond coverage
distribution widens over time when Smp* is dominant
(Fig. b). Using F‾mp=171 W m-2 and other parameters the same as above, we get
Smp*≈0.07 month-1. Although in this case, melting
ponded ice affects pond evolution less than bare ice melting, it can become
stronger if the pond coverage is higher. For example, Smp* and
Sbi* are roughly the same at x=0.35, while at x=0.5
Smp* is roughly twice as large as Sbi*.
The parameter Sbot* controls pond growth by freeboard sinking
due to melting of the ice bottom. The pond growth rate due to bottom melting
increases with increasing melt pond fraction, although more gradually than in
the ponded ice melting case (Fig. , red line). Since the
growth rate is proportional to the bare ice fraction, Sbot*∝(1-xi), the pond coverage distribution gets concentrated over
time (Fig. c). Using F‾bot=20 W m-2 and other parameters the same as above, we get
Sbot*≈0.04 month-1. The contribution from ice
bottom melting becomes larger than the contribution from bare ice melting
only at high x.
Now, we will turn to enhanced melting. The parameter Sem*
controls pond growth by enhanced melting and is the least constrained in our
model due to the many poorly constrained physical processes that potentially
contribute to it. Here we will only consider enhanced melting due to
height-dependent processes (Eq. 24) and leave lateral melting for
the discussion (Sect. ).
Because the growth rate by enhanced melting is inversely proportional to the
hypsographic curve, pond growth by enhanced melting is very fast at the
beginning of the melt and decelerates afterwards (Fig. , cyan
line). The enhanced melting strength is inversely proportional to the square
of the ice thickness, Sem*∝1H2, meaning that
it is significantly more sensitive to variations in thickness than freeboard
sinking. However, it is significantly less sensitive to variations in ice
roughness (Eq. 29). Even on perfectly smooth ice, σ^=0, ponds
will grow due to enhanced melting. In that case, however, lateral melt,
rather than height-dependent enhanced melting, may dominate.
The strength of enhanced melting is proportional to the height below which
enhanced melting is operational, Sem*∝Δs. If we
take ice wetting as a physical example, this means that enhanced melting is
sensitive to microphysical processes that determine how high above sea level
the ice will be wet. The dependence on the parameter k depends on its
magnitude. It appears in Sem* in the term
k-1dsem/dsfs+1. The term
dsem/dsfs is proportional to k-1.
Therefore, if dsem/dsfs≪1, enhanced
melting is proportional to k-1. However, if
dsem/dsfs≫1, enhanced melting becomes
independent of k. Using default parameters, we find this transition happens
at around k≈1.2. In the example of ice wetting, this means that
enhanced melting is sensitive to albedo variations near sea level when ice
near sea level has a similar albedo to the rest of the floe. However, if the
albedo near sea level is significantly lower than the average, pond growth is
insensitive to variations in properties of ice near sea level.
Enhanced melting is proportional to the cube of the bare ice fraction,
Sem*∝(1-xi)3, making it very sensitive to variations
in initial pond coverage. For this reason, the pond coverage distribution
gets quickly concentrated (Fig. d), and it is possible for
initially less ponded floes to overtake initially more ponded floes. If we
assume ice wetting is the only physical process responsible for enhanced
melting, we can place a rough estimate on Ssm*. Taking k=1.7, Δs=0.06 m, and t=30 days, we get for default parameters
Sem*≈0.31 month-1. This suggests that the
contribution to mean pond coverage from enhanced melting is slightly larger
than the contribution from freeboard sinking after 30 days of melt.
The black line in Fig. shows the total pond evolution using
the default physical parameters. The pond growth rate when both freeboard
sinking and enhanced melting occur is not simply a sum of the growth rates of
the four modes since the equations for freeboard sinking and enhanced melting
are solved separate of each other. Therefore, the dependence of growth rate
on pond coverage (Fig. a, black line) was obtained by finding
the derivative of the pond evolution curve. The pond growth rate first
decreases with pond fraction, indicating that enhanced melting dominates early
in the season and then increases, indicating that freeboard sinking dominates
later in the season. The pond coverage distribution using realistic
parameters narrows with time (Fig. e). Since each growth
mode affects the pond coverage distribution in a distinct way, fitting both
the evolution of the mean and the standard deviation of the pond coverage
distribution in observational data could add constraints on the relevant
strengths. Using the above values of strengths, we find that after a month of
growth bare ice melting contributes to roughly 25 % of mean pond
coverage, ponded ice melting contributes to around 13 %, ice bottom
melting contributes to around 7 %, and enhanced melting contributes to
roughly 55 %.
The red curve is the results of . The black
curve is the solution to Eq. () with F‾lat=KlatF‾mp. The pond albedo and the shortwave,
longwave, sensible, and latent heat fluxes used to find
F‾mp are the same as used in
and Klat=1.5. A nearly perfect agreement between the two curves
suggests that a single nondimensional constant, Klat, is enough
to describe pond growth by lateral melting, and the complicated physics of
lateral melting are important only in determining the value of
Klat.
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Discussion
Lateral melting of pond walls by pond water
In our model, we focused on vertical changes in topography, and neglected
pond growth by lateral melting of pond sidewalls by pond water. We will now
briefly discuss this possibility.
This type of melt was the main focus of , who
carefully calculated the lateral melt rates of pond sidewalls by pond water.
The red line in Fig. shows their results. The rate of change
of pond fraction due to a lateral melt flux F‾lat is
dxlatdt=PAF‾latlρb,
where P is the total perimeter of the ponds and A is the area of the
floe. If F‾lat is constant and the dependence of P on
pond fraction is weak, pond growth is linear, which explains the roughly
linear pond coverage evolution in . In
Fig. , black line, we solve Eq. () assuming a lateral
melt flux proportional to the ponded ice melting flux,
F‾lat=KlatF‾mp, where
Klat is a constant. We use the same energy fluxes used by
and estimate PA≈0.1 m-1
from the aerial photographs taken during SHEBA. A nearly perfect match is
obtained with Klat=1.5. Therefore, a single constant that
relates the rate of melt of ponded ice to the rate of melt of pond walls,
Klat, is enough to capture the effects of lateral melting on pond
growth. This suggests that the complicated physics of lateral melting can, to
a large extent, be ignored. More work would, however, be needed to determine
to what degree Klat varies under different circumstances.
If we ignore the topographic variation above sea level, pond growth due to
enhanced melting also becomes linear (Eq. ). Therefore, lateral
melting can approximately be considered a contribution to enhanced melting,
Sem, although it scales differently with physical parameters than
the height-dependent enhanced melting (Eq. 24). It is important
to note that in this model lateral melt does not depend on ice thickness,
H, or on initial pond coverage, xi, although, in reality, it may depend
on these to some degree. For this reason, the pond coverage distribution
width does not change in time, while the mean increases linearly
(Fig. f).
It is not simple to understand the contribution of lateral melting to pond
growth when both lateral and vertical melting occur simultaneously. Each
point along the pond boundary can expand either by lateral melting or by
vertical melting, but not by both. This is because when a point along the
pond boundary melts laterally, it creates a completely vertical slope at that
point. Therefore a small vertical shift will not grow the ponds, and a large
vertical shift will outgrow the lateral expansion. Therefore, if pond growth
due to vertical melting is strong, the contribution from lateral melting will
be small. This is consistent with observations of and
, who found that on first-year ice the contribution from
lateral melting is small. However, steep topography on level
multiyear ice inhibits pond expansion through vertical motion and could lead
to lateral melting being the dominant mode of growth. This is consistent with
our findings of a large contribution from enhanced melting to pond growth on
multiyear ice during SHEBA (Fig. ).
Effects of density variations and internal melt
So far, we have assumed that all the melt occurs on either the top or the
bottom surface of the ice. However, some of the melt can happen internally,
in the bulk of the ice. Internal melt occurs when trapped brine pockets with
high salt content expand and dilute in order to reach a thermodynamic
equilibrium with the surrounding ice. This phenomenon has been reported to
occur both above and below sea level. Internal melt leads to a reduction in
bulk ice density, ρb, which in turn affects pond evolution.
Accounting for internal melt correctly can be quite challenging as it
requires detailed knowledge of the vertical structure of internal melt and
bulk density. Nevertheless, we find that although the effects of internal
melt and density variation may be significant when considered individually,
if considered together they are likely small.
If internal melt is uniform throughout the vertical ice column, the only
effect is a gradual reduction in ρb over the course of the melt
season, slightly increasing the pond growth rate. If, however,
internal melt has a vertical structure, it will create a vertically
nonuniform bulk ice density which can have more complicated effects on pond
evolution. Variations in bulk density and internal melt affect pond evolution
in the following ways: (1) mass transported across sea level due to rigid
body movement depends on the bulk density at sea level; (2) the volume of ice
removed by local melt depends on the bulk ice density at the surface;
(3) freeboard height depends on average bulk densities above and below sea
level; and (4) internal melt induces rigid body motion by melting mass above
and below sea level, without changing the ice surface. We outline the
procedure to include these effects in the pond evolution model in Appendix D.
The resulting equation for pond coverage evolution has the same form as
Eq. (), with only the strengths modified. Here, we only
qualitatively discuss our findings. Pond evolution is most sensitive to the
following:
The difference between the internal melt rate above and below sea level,
e‾asl-e‾bsl, creating a rigid
body motion. Here, e‾asl/bsl is the energy density used
for internal melting, averaged over all ice above or below sea level. More
internal melt above (below) the sea level will create an upward (downward)
rigid body motion of the floe, slowing down (speeding up) pond growth.
The difference between the bulk ice density at the surface and the bulk
ice density at sea level, ρb(h)-ρb(0), changing the
ratio of topographic change due to local melt to rigid body motion. Using
default parameters, rigid body motion is upwards, slowing down pond growth.
Therefore, a lower (higher) bulk ice density at the surface relative to sea
level increases (decreases) the rate of local melt relative to rigid body
motion, speeding up (slowing down) pond growth.
If considered as independent processes, vertical variations in bulk ice
density and internal melt can significantly alter the rate of pond growth.
For example, assuming ρb(0)=850 kg m-3,
ρb(h)=750 kg m-3, and no internal melt leads to a
roughly 60 % increase in the pond growth rate. However, these processes
depend on each other and have the opposite effects on pond evolution. For
example, a high rate of internal melt above sea level, slowing down pond
growth, will lower the bulk ice density above sea level, speeding up pond
growth.
Density and internal melt can be related via a differential equation,
∂ρb(z)∂t=-e(z)l-∂ρb(z)∂zdsrigid
bodydt, where z is a vertical coordinate within the ice
column. Assuming vertically uniform rates of internal melt above and below
sea level, an approximate long-time solution to this equation yields a
vertically uniform bulk density below sea level, and a linearly decreasing
bulk density above sea level. This also defines a long-time relationship
between the vertical profiles of internal melt and bulk ice density,
e‾asl-e‾bsl=lhdsrigid bodydt(ρb(0)-ρb(h)). Using densities from the example in the paragraph above,
and the rate of internal melt obtained in this way, leads to a roughly
10 % increase in pond growth rate, significantly less than the 60 % we
found when considering only the effects of vertical density structure.
A long-time effect of vertically nonuniform internal melt and density is
always a significant compensation between the two, although there may be
transient effects. For this reason, we believe that including a vertical
structure of density or internal melt in the simple model of pond evolution
model is most likely unnecessary.
Under certain conditions, ponds can stop growing
Here, we will entertain the possibility of pond growth by vertical motion of
the topography stopping entirely for a period of time. This is an example of
a possible transient effect of internal melting, which, although interesting,
seems unlikely.
If there is enough mass removed above sea level to induce an upward rigid
body motion that is able to compensate for the effects of local melting near
the sea level, points near the sea level would move upwards,
dsdt>0, and pond growth would stop. This could,
for example, occur if there is strong internal melting above sea level. After
a time, however, high internal melt above sea level would lower the bulk ice
density at the surface, thereby increasing the rate of local melt and
reinitializing pond growth.
We will use an equation for dsdt that includes the
effects of vertically nonuniform internal melt and bulk ice density we
derive in Appendix D, Eq. (). Requiring that
dsdt(x)>0 for any x, we find the condition for
pond growth stopping as
k<ρb(h)ρb(0)ρiρw1+hF‾bie‾asl-e‾bslρ‾aslρ‾bsl-ρw-ρiρwF‾botF‾bi,
where ρ‾asl/bsl is the average bulk density above
and below sea level. Using the values of internal melt and bulk densities
from the previous chapter and taking ρ‾aslρ‾bsl≈1, we find that in order for ponds
to stop growing, k has to be less than 0.85. This is unlikely as ice near
the sea level likely melts faster than ice higher up. Nevertheless, if
internal melt has not had enough time to adjust densities above and below sea
level, it is possible that pond growth could be stopped for a time by the
action of internal melt above sea level. For example, assuming the same
internal melt as in the previous example but a uniform bulk ice density
(ρb(h)=ρb(0)), pond growth would be stopped at k=1. In this case it is likely that growth by lateral melt would take over, as
Eq. () ensures only that pond growth by vertical motions is
prevented.