This is a quantitative question. The problem is inspired by this event:
On August 5, 2010, an enormous chunk of ice, roughly 97 square miles (251 square kilometers) in size, broke off the Petermann Glacier along the northwestern coast of Greenland. The Petermann Glacier lost about one-quarter of its 70-kilometer- (40-mile-) long floating ice shelf, said researchers who analyzed the satellite data at the University of Delaware.
Question:
Imagine an iceberg that is moving freely in the ocean. Given that the temperature of the surrounding water is $T = 4$ Celsius and the temperature $T=0$ Celsius is evenly distributed throughout the volume of the iceberg estimate how long does it take the iceberg to melt completely in the ocean?
We will find the mass of the iceberg from the event description. Average thickness of the chunc is estimated about $500$ m. For simplify we suppose that the iceberg is spherical during the melting.
Answer
Heat leaves through the surface of the berg $\propto r^2$, but the heat required to melt depends on mass $\propto r^3$. Thus, $t \propto r$. Also assume $t \propto 1/\Delta T$.
I took a cube with $r \approx 2 cm$ and put in in my water bottle $\Delta T = 20 C$. It melted in $5$ minutes. Let $r_{berg} = (8 km * 8 km * 250m)^{1/3} = 2*10^3 m$
$t_{berg} = 3 min 10^5*20/4 = 1.5 * 10^6 min = 3 years$
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