Convection and sea ice formation: An evaluation of three 1-D models

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Polar Oceanography Report (GEOF 338)
Victor Aguiar
Geophysical Insitute, University of Bergen, March 2020
Abstract
The sea ice growth due to water surface cooling is accompanied by brine rejection
into the oceanic mixed layer. Such haline plume, usually heavier than the sorrounding
waters, may trigger convection and hence vertical mixing. In the present work, I
investigate three distinct 1-D sea ice growth models and how different temperature
and salinity initial profiles produce distinct outputs. One of the model, the classic
analytical Stefan’s Law, does not take into account the heat content within the water
column whereas the other two models, Zubov-Defant and Ball-Kraus-Turner, provide
convection depth estimations and an overview of sea ice growth response due to vertical
mixing. Results indicate that a snow layer on the top of the sea ice may indeed act as
a thermal isulator, slowing down sea ice growth and vertical mixing. Moreover, initial
profile stability and mixed layer characteristics seem to play a relative major and minor
role on sea ice growth, respectively.
1 Introduction
Convection is considered a key process in many ocean thermodynamic phenomena such as
mode water formation (Hanawa & Talley, 2001), deep water formation (Rudels & Quadfasel,
1991) and sea ice increase and decay (Solomon, 1973; Fichefet & Maqueda, 1999). Sea ice
covers about 7% of the world ocean (Leppäranta, 1993) but its albedo and insulation charac-
teristics are key components on the earth heat budget (Zheng et al., 2019). Observed sea ice
retreat and thinning (Wadhams & Davis, 2000) and albedo decrease (Pistone et al., 2014)
expose the darker sea surface. Once exposed, ocean-atmosphere heat fluxes are enhanced,
altering the vertical mixing-sea ice growth feedback (Liang & Losch, 2018) and ultimately
leading to the so-called Arctic amplification (Serreze & Barry, 2011).
In this report I will investigate the evolution of sea ice growth, convection depth and
stemperature (T) and salinity (S) by using three distinct 1-D models: Stefan’s Law, Zubov-
Defant model (ZD) and Ball-Kraus-Turner model (BKT). The document is structured as
follows: Section 2 briefly describes the sea ice growth and convection theories, the applied
models and the considered initial profiles. Section 3 presents the results obtained by using the
different models and profiles. Section 4 contains the discussion about the achieved outputs.
1
2 Theory and Methods
2.1 Basic Equations
The initial equations that one must take into account is the well known heat conduction :
∂
∂t
(ρiciT ) = ∇ · (κi∇T ) + q (1)
where t is time, ρi is the ice density, ci is the specific heat of ice, T is the ice temperature,
κi is the heat conductivity of ice and q is an internal source term (Leppäranta, 1993).
The first simplification considered here is that vertical temperature gradients are usually
much larger than the horizontal ones. In other words, Eq. 1 may be rewritten as:
∂
∂t
(ρiciT ) =
∂
∂z
(κi
∂T
∂z
) + q (2)
Two boundary conditions are applied: one at the top, determined by the heat flux (QT )
at the surface, and the other at the bottom, where the temperature is fixed to the freezing
temperature of the sea water (Ts).
top : κi
∂T
∂z
= QT (3a)
bottom : T = Tf (3b)
However, the lower boundary level changes due to freezing and melting:
ρiL
dH
dt
= κi
∂T
∂z
|bottom −Qw (4)
where L is the latent heat of freezing, H is the ice thickness and Qw is the heat flux from
the water to the ice.
This set of equations (2)-(4) describe the evolution of sea ice thickness and heat conduc-
tion through ice.
2.2 Stefan’s Law
The Steffan’s Law is obtained when the following assumptions are applied on the Eqs. 2 - 4
1. No thermal inertia
2. No internal heat source
3. A known temperature at the top, T0 = T0(t)
4. No heat flux from the water
2
The first and the second assumptions simplify the Equation 2 to a linear temperature
profile within the ice. The third assumption simplifies the top boundary condition by defining
a known temperature time series. Finally, the last assumption (Qw = 0) simplifies the
Equation 4 to:
ρiL
dH
dt
= κi
Tf − T0
H
(5)
One can find the analytic solution, the Stefan’s Law, by time integrating Equation (5).
Thus:
H =
√
2κi(Tf − T0)
Lρi
∗ t (6)
Leppäranta (1993) points out that the simplifications assumed to derive Stefan’s Law
tend to positive bias, i.e., the solution tends to overestimate the sea ice growth. As an
example, the maximum observed sea ice thickness of underformed ice in the Baltic Sea was
122 cm while Stefan’s Law estimate it in 135 cm. I encourage the motivated reader to check
the detailed explaination provided by the aforementioned author regarding the consequences
of the considered simplifications.
As one may naturally suppose, the addition of snow on the preceding system implies on
introducing new modifications on the equations and challenges regarding the modeling task.
First, due to its low heat conudctivity, the snow layer can strongly insulates the sea ice thus
reducing the basal growth (Zhao et al., 2019). Secondly, the weight of the snowpack may
create a negative freeboard, i.e., the ice is completely submerged and snow slush might form.
The latter is not represented on the model.
Lastly, this analytical solution does not solve the convection depth once that ocean and
ice systems are not coupled.
2.3 Freezing and Brine Release
As the seawater freezes, salt is rejected from the ice cristal to the water as brine solution.
It is well known that the density of seawater (ρw) is a function of temperature (T ), salinity
(S ) and pressure (P), i.e., ρw = ρw(S,T,P) and hence brine release will increase the density
on the region of sea ice growth.
2.4 Convection and Convection Depth
A well mixed layer is present between the growing ice and a strong pycnocline. As mentioned
previously, sea ice growth releases a cold a salty brine solution and such plume, heavier
than the sorrounding mixed layer waters, will become gravitationally unstable and induce
a vertical circulation. Known as haline convection, such downward motion will increase the
salinity, and thereby the density, of the layer by turbulent vertical mixing. The depth at
which the haline densier plume reaches its neutral buoyant point, thus achieving gravitational
stability, is known as convection depth.
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2.5 1-D Models
The three sections above tell us a somehow intuitive story of the simplified physics behind
sea ice formation. At the same time, even though in a subtle way, these same equations
introduce the idea of applying numerical models in order to predict and discuss sea ice
growth, convection depth and T-S time evolution. These are in fact our ultimate goal.
Until around 1970 Zubov (1943) and Defant (1949) models were considered the standard
approach for treating unstable surface convection (Solomon, 1969). Developed and pub-
lished independently by the authors, in this model salt released due to sea ice formation
mixes throughout the entire mixed layer increasing its salinity and thickness until its density
becomes equal to that of the bottom layer. However, the water below the mixed layer is not
affected by the mixing process, in other words, it is assumed to be completely undisturbed.
Then, the two layers mix and become the new surface mixed layer. The process is repeated
sequentially (Solomon, 1969, 1973).
The oversimplifications of the Zubov-Defant model started to get evident soon after
their publication and perhaps its most deficient characteristic is the inhability of efficiently
eroding the pycnocline (Solomon, 1969, 1973). Using data from ice island Arlis II, Fujino
(1966) found that the increase of salt released due to ice formation predicted by the ZD
model is lower than the increase of salt observed in the mixed layer. In this sense, Fujino
(1966) suggested that such difference might be related to the entrainment of saltier water
from below into the mixed layer. Moreover, another weakness of thethe ZD model is that
it does not allow a layer of snow over the ice.
A series of experiments and observations, initiated by Rouse & Dodu (1955), indicate
that penetrative convection might be the mechanism responsible for steeping the pynocline
(Solomon, 1973). Ball (1960) observed that the turbulent upward motion, associated with
thermal convection, is not destroyed by neither molecular nor eddy viscosity within the
mixed layer. Instead, it is primarily destroyed by entrainment of warmer air from above.
Although Solomon (1969, 1973) do not clarify the meaning of ’steep pycnocline/halocline’ I
understand that as they are remarkably eroded, as an indication of penetrative convection.
Later, Kraus & Turner (1967) demonstrated in an experimental study that the poten-
tial energy of the plume is first converted into kinematic energy, it penetrates beyond the
equilibrium level promoting entrainment from the water below into the upper layer. It is
assumed that the potential energy of the system is conserved.
2.6 Scenarios
Four distinct scenarios – or initial profiles – will be considered on this report and all of them
are suggested in the guideline sheet provided to the students. Furthermore, all experiments
are conducted with constant air temperature of -15oC, constant sea ice heat conductivity
(κi = 2 W m
−1 K−1) and from 01/10/2015 to 01/05/2016. The first initial profile is based
on the observed values of temperature and salinity in Billefjorden, in Svalbard. The second
experiment also considers the same T-S profiles but it also takes into account a snow cover of
10 cm with heat conductivity κs = 0.15 W m
−1 K−1. For the third experiment, I will consider
a scenario similar to a freshwater lake, with the temperature profile being the Billefjorden
one but with constant salinity S = 0. Finally, the last experiment considers a initial profile
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mixed over the first 65 meters by a storm.
3 Model Results
In this section I will present the results for the four experiments defined above.
3.1 First Experiment
Figure 1 shows the evolution of temperature, salinity and density profiles (upper panel) as
well as the ice thickness and convection depth (lower panel). From this Figure it is clear
that the ice thickness produced by the Stefan’s Law solution (approx. 1.8 m) is greater than
the values obtained by using the ZD (1.3 m) and BKT (1.6 m) models. Another remarkable
feature observable in Figure 1 is how the ice thickness provided by ZD increases faster than
the BKT in the beginning but is overtaken around mid-January. Furthermore, water reached
the freezing point 5 days after the start.
The convection depths reached by ZD and BKT also reveal distinguishable features.
While in the latter it increases faster and reaches a maximum (175 m) after 3 months, the
former presents a continuous growth until the end of the run, reaching only 92 m. It is
worth pointing that the sea ice thickness curve produced by the BKT model overtakes the
ZD curve a few days earlier than the former reaches its maximum convection depth.
With regards the vertical profiles, one may clearly observe how the density penetrates
further down in the BKT model in comparison to the ZD model. It must also be highlighted
that, in the case of the BKT outputs, the water column is completely mixed 2162 hours (
90 days) after the start.
3.2 Second Experiment
Results of experiment two, the one with the same T-S initial profiles but now also considering
a snow layer of 10 cm and κs = 0.15, are shown in Figure 2. Differently from the first
experiment, BKT profiles did not evolve to a fully mixed state. One may clearly observe
such difference when comparing top panels of Figure 1 and Figure 2.
The ice thickness and convection depth evolution also present significant changes in
experiment 2. After the 6 months run, the final ice thickness is about 1 m for the Stefan’s
Law model whereas it reduces to less than 0.2 m for BKT model. It is interesting to notice
that no ice was formed during the first two months with BKT. Lastly, the convection depth
also presented a distinguishable decrease, reaching only 75 m at the end of the run. One may
observe that at the beginning the convection depth quickly reached 50 m, converging to the
observed result in experiment 1. However, differently from this last one, the convection depth
remained constant until the end of the second month (approx. 1500 hours) and smoothly
deepened as time evolved. Again, the ice thickness increase seems to follow the convection
depth evolution, both of them presenting a more visible concomitant increase after the 1500
hours mark.
5
(a) (b)
(c)
Figure 1: (a): Time evolution of salinity, tempererature and density profiles produced by
the BKT model. The period between two lines is 15 days. (b) Same as (a) but produced
by the ZD model. (c) Ice thickness evolution provided by the Stefan’s Law, ZD and BKT
models (solid red-ish lines) and convection depth for ZD and BKT (dashed blue-ish lines).
3.3 Third Experiment
One point must be elucidated before presenting the results of this experiment. At first, my
plan was to maintain the temperature profile as it was provided to the students, i.e. without
alterating it, but change the salinity profile to a constant value of S = 0. However, I was
getting an error when running the model. This error, related specifically to the BKT model,
might be due to the fact that the initial conditions are now unstable. In this sense, in order
to obtain a stable initial condition, I multiplied the temperature profile by -1, considered the
salinity profile constant (S = 0) and the results related to this configuration are presented
in Figure 3.
Both T-S evolution (Figure 3 (a)) and convection depth and sea ice thickness (Figure
3 (b)) are remarkably different from the other 3 runs. First of all, the former present a
stagnant step-like structure with non-realistic density values. Secondly, the three models
presented exactly the same ice thickness development, reaching more than 2 meters of ice
thickness at the end of the run, with BKT and ZD presenting the same constant convection
depth (10 m) throughout the run after only 18 hours of freezing.
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(a) (b)
Figure 2: (a): Time evolution of salinity, tempererature and density profiles produced bythe
BKT model. The period between two lines is 15 days. (b): Ice thickness evolution provided
by the Stefan’s Law and BKT models (solid red-ish lines) and convection depth for BKT
(dashed blue-ish lines).
(a) (b)
Figure 3: (a): Time evolution of salinity, tempererature and density profiles produced by
the BKT model. (b): Ice thickness evolution provided by the Stefan’s Law, BKT and ZD
models (solid red-ish lines) and convection depth for BKT and ZD (dashed blue-ish lines).
The obtained results are the same so lines are overlapping each other.
3.4 Fourth Experiment
The fourth and last experiment simulates the situation of which a storm mixes the first 65
m of the water column. The T-S values of temperature and salinity after the virtual mixing
are:
Depth [m] Mean T [oC] Mean S
Before 19 -1.038 33.83
Later 65 -0.396 34.05
One may notice that the results shown in Figure 4 present similar features as the ones
obtained in the first experiment (see Figure 1). Nonetheless, differences are also present and
the first one to be highlighted is the amount of time for the surface water to reach the freezing
point. While in the first experiment it took 5 days, in this one it was 17 days (≈ 412 hrs).
Furthermore, the BKT convection depth in this experiment reached the maximum value in
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76 days (≈ 1842 hrs), earlier than in experiment 1 (90 days). The final ZD convection depth
was 107 m, approximately 15 meters deeper than in experiment 1.
Such differences in the convection depth enhanced vertical mixing and as a result it
prolongated the freezing window, culminating in further ice growth development. Even
though the sea ice thickness for the Stefan’s Law case was the same as in experiment 1, for
BKT and ZDit increased for approximately 1.7 m and 1.45 m, respectively.
Figure 4: Ice thickness evolution provided by the Stefan’s Law, ZD and BKT models (solid
red-ish lines) and convection depth for ZD and BKT models (dashed blue-ish lines).
4 Discussion
As observed by Leppäranta (1993), and also found in this report (see Fig. 1), the ice thickness
generated by Stefan’s Law overestimate the sea ice thickness, although its shape presents
the same pattern as the curves produced by the ZD and BKT models.
Perhaps the greatest discrepacy, or at least the most visible difference between ZD and
BKT model outputs in experiment 1, relates to the convection depth. As mentioned previ-
ously, these two models fundamentally diverge with respect how they treat haline convection
(see Section ”1-D Models”). In this sense, due to the penetrative convection and entrainment
from waters from below into the mixed layer, the convection depth in the BKT model grows
much faster and reaches higher depths, the whole water column in fact, than the ZD model.
After reaching this point, the entire water column is mixed and thus ice formation speeds
up, overtaken the curve produced by the ZD model.
Moreover, Solomon (1973) points out that in the absence of mechanical stirring, one
could expect that the actual extent of the penetrative plume falls between the BKT and ZD
model.
As mentioned previously, snow acts as a good thermal insulator and this feature is well
depicted in Figure 2. Due to its low thermal conductivity (κs << κi), heat transfer from the
ice-ocean to the atmosphere is reduced, slowing down basal ice growth (Fichefet & Maqueda,
8
1999). As a consequence, vertical mixing is also inhibited and both processes present a more
clear increase just after the convection depth 50 m barrier is overtaken. It is clear from the
top panel that the initial temperature profile presents a peak at about 50 m and even though
it was rapidly eroded on the first experiment, it actually acted as a thermal barrier on the
second experiment. Once the convection mixed this layer, heat transfer was improved and
hence ice thickness could develop.
The two last experiments highlighted the importance of the vertical termohaline structure
on sea ice growth. The third case simulates a thermally stratified lake with constant S = 0.
Although the results presented in Figure 3 seem to be somehow cofusing at a first glance,they
apparently describe a situation of instability followed by a very stable column. Firstly, the
upper 10 m layer is rapidly cooled to the freezing point and this water, denser than the lower
layer, falls until its stability depth at about 75 meters where an even denser layer relies under
that. Once that the top layer is at the freezing point and there is no brine release, vertical
mixing due to the latter is inhibited. Hence, the heat content within the water column is not
made available and the sea ice growth happens solely because of the difference between the
sea surface and asmophere temperatures, thus leading to a exactly convergence of models’
outputs.
Finally, the last experiment reavealed the role of the mixed layer in sea ice growth. The
first 65 m of the T-S profiles were virtually mixed, hence changing - in this case increasing -
the values of both properties in the new mixed layer (see Table 1). As a result, now warmer
and thicker, freezing occurred about 12 days later than in the first experiment. The additional
heat came from the aforementioned thermal barrier. Present at something between 50 m
and 75 m depth, it was partially absorbed in the virtually mixed top layer. After reaching
the freezing point, convection erodes the now thinner thermal boundary more easily than in
the first experiment. As a result, although freezing was delayed, convection depth maximum
value was reached earlier with BKT and its magnitude with ZD was greater when both are
compared to experiment 1, implying in an increase of the ice growth period window and thus
allowing a slight increase of sea ice thickness (6.25% and 11.5%, respectively) until the end
of the run. It must be noticed that Stefan’s Law model results are not as much modified as
ZD and BKT because it does not consider the heat content within the water column.
5 References
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