Increased human activities in the Arctic has led to the diminishment of Arctic sea ice, about 70,000 km2 per year and has raised concerns for the region’s future. Measurements show that the ice has grown thinner, approximately 40% in the last two decades. The region is opened to increased human activities like commercial shipping, Arctic oil and gas exploration, in addition to deposition of soot by the maritime vessels. Black carbon from incomplete combustion is lodging over the ice and is causing graying of ice caps which was once a reflective surface to absorb more of sunlight and warm the water. Increased water temperatures are having grave impacts on the flora and fauna that are dependent on ice. In near future Polar bears are likely to face extinction as their breeding habitat is given to melting ice. Trapped green house gases like methane are released due to the melting areas of permafrost. Some simple maths can give us the glimpse of the complexity faced by the scientists in handling ice-ocean-climate models.

1. The Melting North Pole
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2. Table of Contents
Introduction...................................................................................................................................................3
Growing of ice in North Pole:.......................................................................................................................3
Moving of ice:...............................................................................................................................................5
Statistics........................................................................................................................................................6
Conclusion ....................................................................................................................................................7

3. Introduction
Increased human activities in the Arctic has led to the diminishment of Arctic sea ice, about
70,000 km2
per year and has raised concerns for the region’s future. Measurements show that the
ice has grown thinner, approximately 40% in the last two decades. The region is opened to
increased human activities like commercial shipping, Arctic oil and gas exploration, in addition
to deposition of soot by the maritime vessels. Black carbon from incomplete combustion is
lodging over the ice and is causing graying of ice caps which was once a reflective surface to
absorb more of sunlight and warm the water. Increased water temperatures are having grave
impacts on the flora and fauna that are dependent on ice. In near future Polar bears are likely to
face extinction as their breeding habitat is given to melting ice. Trapped green house gases like
methane are released due to the melting areas of permafrost. Some simple maths can give us the
glimpse of the complexity faced by the scientists in handling ice-ocean-climate models.
Melting of ice in North Pole:
In the present times the melting of ice in the North Pole has widely increase. This is harmful for the
ecosystem as there is a sudden extinction of many species such as polar bear and plankton. Also, it leads
to the release of the gases such as methane, the effect of the same has already been observed in the
Pacific Ocean.
Air and water temperature, wind and ocean currents highly influence the sea ice. We are landed
up with some questions like, How fast the ice grows?, What can be the thickness of the ice?, and
so on. Let’s consider a block of ice of uniform thickness, of surface area about 1m2
is floating on
the water. The ice block has a column of air above it, which has constant temperature below the
freezing point of sea water and the water below the ice is above the freezing point. Fourier’s law

4. of heat conduction can describe the rate of heat transfer, which if given by the following
differential equation
Where, Q is the heat, T is the time, k is the thermal conductivity of the ice, h is the thickness of
the ice , is the temperature at the ice/air interface and is the temperature at ice/water
interface. A layer on the water surface freezes as the water below the ice loses heat. This
formation of ice can be explained by the following relation between the mass m of the layer and
the heat loss Q:
Q=Lm, where L is the latent heat.
The product of density D and volume of water gives the mass m of the layer of water. Density is
proportional to temperature but the temperature of the layer closer to the ice/water interface is
constant(Alexiades.1992). Thus, the loss of heat of water through the ice layer is Q joules, and
the thickness of ice h grows by an amount Q/LD,
On combination with the previous equation, the resulting differential equation is
And the solution to the above equation is the function given as:
where is ice layer’s initial thickness.

5. Rate of growth slows down with the increase in thickness h of the ice. Ice layer of different
thickness can be leveled, as the thin ice grows more rapidly than thick ice. It has been observed
that the equilibrium thickness of ice is 2.88 meters.
Moving of ice:
To one’s surprise ice moves in a constant motion. It is driven by the ocean currents and the
winds that tear at their surfaces. Cracks and ridges develops due to the pressure and stress
created by the wind and the ocean currents, and thus they change their response according to
wind and ocean. To understand this movement and its dependence on external forces in an
efficient manner, we use Newton’s second law of motion, which states that the acceleration of
the object equals the result after dividing the net force acting by the mass of the object.
Again, let’s take a block of ice of uniform thickness with area of 1m2
. Resulting force is sum of
various stresses acting on the ice that includes stress both due to wind and water. Coriolis force,
the force exerted on the object by the movement of the earth cannot be eluded, as it plays a vital
role in movement of the ice. The above stated words can be made clear by the following
expression
which can be written as
, , and are vectors describes both the motion and magnitude of the force.

6. Wind exerts stress on the ice and drags it along in the flowing water, which is calculated using
the equation where is the density of the fluid, A is body’s surface area and
is water’s velocity. C is the constant defined as the drag coefficient of the body and it is also
responsible for the shape of the object body. Object with bulky body provides greater resistance
to the flowing water than the thin and flat one. The amount of drag increases as the square of the
airflow’s speed. Based on assumption, value of A is taken as 1 and the value of C is between
0.0014-0.0021.
If the ice flow is not in motion, the equation takes the following form
where is the air velocity. The relative velocity to be used is ,
taking the self-movement velocity of ice into consideration. Thus, the equation takes the
form:
In similar fashion, we can determine the stress exerted by water using water density along-with
an underwater drag coefficient. The expression for Coriolis force is given as:
where, m is the mass of the ice-piece, velocity of ice, is latitude and
angular velocity of the earth (Golden & Perovich, 2014). The direction of the force is 90 degrees
to the right of the velocity of the ice in northern hemisphere and 90 degrees to the left of the
velocity of the ice in southern hemisphere.
Statistics

7. Another vital feature of ice-ocean modeling is statistics. All mathematical and experimental
models lay their emphasis on the observations and their predictions are compared with real
values. Collecting the information is not a child’s play. Satellites gives the pictures of the surface
of the ice whereas submarines gives the picture of two-dimensional vertical slice of ice.
Conclusion
The Arctic has a complex environment and is largely uncharted. International customary
maritime law-UNCLOS governs the arctic and Arctic council is responsible for the cooperation.
EU also seconds the multilateral approach. The point at which ice-capable technology,
delineation of maritime and continental shelf boundaries, high energy prices, climate conditions,
and proper management of maritime traffic converges in sufficient manner, then only prospect
for investment and utilization can be undertaken (Stroeve et al, 2014). The six initiatives of
January 2014 Implementation Plan for National Strategy for Arctic region includes increasing
awareness of arctic domain, preventing spilling of hazardous material, containment, enhancing
arctic search and rescue for the same and promoting arctic waterways management. Arctic
exploration is not the matter for the future, the matter to be concerned is the geopolitical
importance which can be gained when the oil prices increases, Ice-capable technology will have
to be available throughout, despite seasonal or year-to-year fluctuations, interests in maritime
shipping routes must hike and geopolitical community should be convinced that any commercial
development will not hamper the Arctic environment. It’s high time for everyone to set in action
now. Let’s get on with our duty towards the environment and take every step to save the Arctic.
References
Alexiades, V. (1992). Mathematical modeling of melting and freezing processes. CRC Press.

8. Golden, K. M., & Perovich, D. K. (2013). Multiscale Models of Melting Arctic Sea Ice. UTAH UNIV SALT
LAKE CITY DEPT OF MATHEMATICS.
Stroeve, J. C., Markus, T., Boisvert, L., Miller, J., & Barrett, A. (2014). Changes in Arctic melt season and
implications for sea ice loss. Geophysical Research Letters, 41(4), 1216-1225.