Covering 7-10% of the Earth’s surface, Sea ice is a critical component of the climate system and is sensitive to changes in global temperature. In the paper “Nonlinear threshold behavior during the loss of Arctic sea ice” (Eisenmann & Wetlaufer 2009) a low order model for sea ice thickness is presented which exhibits hysteresis in sea ice loss as the climate warms. This model considers the most impactful processes affecting sea ice volume and parameterizes them, good representations of these processes is thusly paramount. One such important process is the ice albedo feedback. In the Arctic summer months, snow melt turns into dark ponds which sit atop the sea ice. Dubbed melt ponds, these ponds lower the over albedo (reflectance) of the ice causing the absorption of more incoming solar radiation, promoting more melting and further lowering the ice albedo and continuing. In this project we will investigate how different parameterizations of this process affect model output and hysteresis. We can also examine how changing seasonal temperature variations, such as early warming and melting, will affect the fate of the Arctic sea ice pack. Group members: Madeleine Braye, Yasmin Eady, Nicole Jacobs, Jesse Liu, Ryan Norlinger, Jose San Martin

1. An investigation of low-order sea ice models
with data driven parameterizations of
important physical processes.
Maddie, Jesse, Jose, Ryan, Yasmin, Nicole

2. Maddie

3. Jose

4. Ryan

5. Nicole

6. Jesse

7. Yasmin

8. Christian

9. Paul

10. How does Arctic Sea ice change?

11. 2 Datasets
 Melt Pond formation in June
 Proportion of Ice, Melt Ponds, and Sea Water from 4 Arctic Sites

12. Arctic Sea Ice Melt Pond Statistics and
Maps
 A GeoTIFF file covering 10km by 10km at 1m resolution
 Analyzed with supervised maximum likelihood classification to
derive pond, open water, and ice surface classes
• 4 Arctic Ocean sites within the median extent of the
perennial ice pack
• Irregularly spaced in time

13. Average Albedo of Arctic Ocean Sites
• The albedo of each map at each date is a weighted average albedo of pond, open water, and ice
• αwater = 0.2
• αice = 0.85
• αpond = 0.25~0.6

14. 3 Stages of Summer Arctic Sea Ice
Melting
Pond
Formation
Ponds drop to sea
level
Ponds lost to ocean
cracks

15. Dealing with Uncertainty in Energy &
Albedo
 𝑬 = 𝑳𝒊 ∗ 𝑻𝒊 ∗ (𝑪𝒊 ∗ 𝑯𝒊 − 𝑪 𝒑 ∗ 𝑯 𝒑)
 𝐿𝑖 = latent heat of fusion of ice
 𝑇𝑖 = temperature of Arctic ice
 𝐶𝑖 = % coverage by Arctic ice
 𝐶 𝑝= % coverage by melt ponds
 𝐻𝑖 = depth of Arctic ice
 𝐻 𝑝= depth of melt ponds
𝐾𝑛𝑜𝑤𝑛
𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑
(Gaussian with mean 7.39cm and std dev of
8cm)
• 𝑨 = 𝑪𝒊 ∗ 𝒂𝒊 + 𝑪 𝒘 ∗ 𝒂 𝒘 + 𝑪 𝒑 ∗ 𝒂 𝒑
• 𝑎𝑖 = albedo of Arctic ice
• 𝑎 𝑤 = albedo of ocean water
• 𝑎 𝑝 = albedo of melt pond
𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑
(uniform between 0.25 ~ 0.6)
𝐾𝑛𝑜𝑤𝑛

17. • Original model has 4 ice-free
solutions
• Model reparameterized on melt
pond data has 7 ice-free solutions

18. Modeling the drainage of melt ponds
 As E increases, the albedo decreases as melt ponds are formed. However,
as it continues to increase, the sea ice becomes porous, allowing melt
ponds above sea level to drain.
 The higher freezing point of fresh water
causes the draining water to create
blockages in the ice, reducing the drainage
of the melt pond between certain ice
temperatures.

19.  The melt pond drainage and resulting blockages cause the albedo of the
ice caps to increase within the temperature bounds. Therefore, instead of
the constant decrease seen in current models, there should be a slight
increase in albedo when the energy of the ice would cause drainage.
 Our goal is to factor in this change in albedo, allowing us to get a better
idea of how these variables will affect the sea ice cover.

20. By changing f(t) to different functions, we can model the melt pond drainage and
create a more accurate model of sea ice coverage as the energy of the system
changes.

21. The graphs on the left are only affecting the albedo within the critical temperature range. In reality, we would
expect the drainage to have more long lasting effects, like in the right hand graphs.

22. Using xpp, we looked at how these changes in
albedo would affect the equilibrium solutions for the
energy of the system. The standard equation yielded
the top graph when ranging the flux from 0 to 100 in
steps of 5. The small bumps yielded no change, but
the longer effects caused the graph to contain one
less equilibrium solution with a negative energy
state.

23. One goal of this project was to investigate if modeling the pond drainage would affect the bifurcation of
the original equation. The piecewise equation used to model the differences is not continuously
differentiable, and therefore the bifurcation does not go continuously, but by approaching the equation
from different initial conditions, we were able to see that the bifurcation persists.

24. To further investigate effects of melt pond drainage, we graphed
both parts of the change in energy equation to see when the
energy is at equilibrium. When the change in flux is 0, the only
equilibrium is at a very low energy, forming a stable solution. As the
imposed heat flux increases, the line shifts down, forming more
solutions as it crosses the curve.

25. These extra equilibrium points create
interesting dynamics. In the model with no
flux, there is only one equilibrium point, which
is at a very low energy. As the flux increases,
more equilibrium points are created by the
line interaction. The introduction of the melt
pond drainage increases extra interactions,
and in a range between 35 and 45 Wm-2 flux,
3 new equilibrium points are created. These
points include a new stability point at a much
more reasonable energy.

26. By looking at a range of fluxes that covers this difference, we
were able to investigate the difference between the standard
equation and the new one incorporating melt pond drainage.
It is evident that the two equations, although they start at the
same value, go to different equilibrium states.

27. Constant Average Solar Energy Input
= [1-a(E)Fs(t)-F0(t)+ΔF0-FT(t)T(t,E)+FB+v0R(-E)]
 Fs(t) – down-welling seasonal radiation
 F0(t) & FT(t) - outgoing longwave radiation
 ΔF0 – proxy for C02
Simplified Model with Average Solar Energy Input
𝒅𝑬
𝒅𝒕
= (1-((.2+ai)/2+((.2-ai)/2)*tanh(E/(9.5*.5))))*100-85+F-
2.8*E/(6.3)+2+sqrt(4)
𝒅𝑬
𝒅𝒕

28. Fourier Series

29. Temperature Dependent Surface Flux
FT(t)
• Fourier series Representation of
Discrete NCAR data as a smooth
curve
• Original Average: 2.8 W/mk
• 𝐹𝑡 brings in the seasonality of arctic
cloud cover and is scaled by ice
temperature.

30. Temperature Independent Surface Flux
F0(t)
• Fourier series Representation of
Discrete NCAR data as a smooth
curve
• Original Average: 85/W2
• 𝐹0 brings in the seasonality of arctic
cloud cover and heat fluxes from
lower latitudes.

31. Incident Shortwave Radiation Flux
FS(t)
• Fourier series Representation
of Discrete NCAR data as a
smooth curve
• Original Average: 100 W/m2
• 𝐹_𝑠 brings in the seasonality
of solar radiation.

32. Original Model
More ice
No
ice/warmer
water

33. Changes in Ice Energy by C02 Levels
More ice
No
ice/warmer
water

34. Future Goals
 𝐹0 takes into account lower latitude temperatures, but in the original model
these were not increased with Co2, we would like to investigate the effect
that warming lower latitudes would have on the model.
 We would also like to investigate how a warmer climate might effect the
cloud cover in the arctic since warmer climates likely have more water
vapor in the atmosphere.
 Optimize albedo parameters for more accurate model inputs.