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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.

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+1 for assuming a spherical iceberg. ;p –  Raskolnikov Dec 15 '10 at 13:36
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+1 for not asking whether the iceberg would move left or right in absence of currents ;-) –  Sklivvz Dec 15 '10 at 15:56
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@Sklivvz If you like riddles, here you are -- one puts an ice cube in a glass and fills it up with water. How much water will flow out when the ice cube melts? –  mbq Dec 15 '10 at 20:47
    
Depends on how big is your thirst. –  Martin Gales Dec 16 '10 at 7:59
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@mbq, that's too easy :-) –  Sklivvz Dec 16 '10 at 14:25
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3 Answers 3

up vote 11 down vote accepted

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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+1; as an addition, a paper with lifetime/size scatterplot with some actual measurements (pdf link): geoanalytics.net/GeoVis08/a21.pdf –  mbq Dec 15 '10 at 13:06
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+1 for actually performing the experiment –  Sklivvz Dec 15 '10 at 15:56
    
I guess that the flatness of the iceberg helps it melting faster, whith a time only proportional to its smallest dimension. Here it would be a factor 250 m / 2 cm = 10⁴, corresponding to 1 or 2 month. –  Frédéric Grosshans Dec 15 '10 at 17:08
    
@Frederic That sounds like a reasonable way of thinking about it. Experiment? –  Mark Eichenlaub Dec 15 '10 at 19:07
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@Martin, Frederic The temperature of the water is included in the estimate - I assumed heat flow is proportional to temperature difference. I don't know how accurate the assumptions are - it was just a back-of-the-envelope guess. –  Mark Eichenlaub Dec 16 '10 at 13:18
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I don't think its very easy to answer. This is a large tabular berg, so its width is many times its height. The key will be fluid flow of warm seawater to its underside. Without this flow, the sea underneath the berg would be depleted of heat and melting would stop. The transport of warmer water is probably determined by a combination of ocean currents and storm driven waves.

Also note: 4C is a max (not min) density for water, and thus would sink. Sea water freezes at roughly 28F (-2C), so if it is embedded in water that is at the feezing point, the heat transfer would be from the iceberg into the sea. {But actually an antarctic iceberg probably has an internal temperature of -20 to -40C}. It probably has to be transported north into warmer ocean water before the melting really gets going.

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This is a very difficult question to answer in detail because:

  • melting produces a layer of cold water around the berg
  • the iceberg changes its geometry while melting
  • the iceberg will roll in response to changes in geometry
  • the iceberg is transported, changing boundary conditions

This is a nonlinear nightmare. In any case, the back-of-the-envelope calculations by Mark are fairly OK and reasonable.

Many years ago there was an special volume of "Annals of Glaciology" dedicated to explore the question whether icebergs could be towed from Antarctica to arid regions in the middle East for irrigation purposes. It is a very enriching volume. Check it out here:

Annals of Glaciology 1

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Thank you for the answer! –  Martin Gales Jun 3 '13 at 15:14
    
You are welcome! –  Cinco Sauces Jun 3 '13 at 15:16
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protected by Qmechanic May 25 '13 at 10:23

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