4. \"Ballpark\" estimates of time scales for melting an ice sheet Even for a com

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4. "Ballpark" estimates of time scales for melting an ice sheet Even for a complicated system like an ice sheet, it can be useful to get a rough idea of the range of possible behavior from the simplest considerations. People often make the mass balance estimate of how much sea level would rise if a certain ice sheet were to melt. But making guesses about the energy budget can give (very rough) estimates of how long it might take to melt. Round your answers to only 2 significant digits, since the inputs given below are only given to 1 or 2 significant digits (a) For instance, suppose you have 9 W/m2 surface energy imbalance from greenhouse gases causing melting on an ice sheet 1.5 km thick (about the average thickness of the Greenland ice sheet). For simplicity, we also suppose the temperature is close enough to the freezing point that all this excess energy goes into melting (as opposed to sublimation or increasing the surface temperature). Using a latent heat of fusion of L 3.3x10 J/kg, and assuming ice density of about 0.9x103 kg/m3, how many m/yr of ice melt, and how many years would it take to entirely melt 1.5 km thick ice just with the flux into its upper surface? [Hint: (ice thickness × density × L) gives energy needed per m2 to melt the ice: divide by the energy imbalance per m to find the time; convert to years. There are 86400 x 365 seconds p er ear (b) Using the answer from Part (a) for the number of m/yr of surface ice melt, and assuming that this happens over 20% of the area of the Greenland ice sheet 0.2"( 1.7x10 km), how large is the contribution to mean sea level rise in cm per decade? (Show your work). The area of the world ocean is 3.6x108 km2 (for this you can assume the added water is evenly distributed, even though 6 AOS102 Problem Set II in practice ocean currents affect the geographical distribution). Is your number roughly consistent with the contribution of land ice discussed in 2c? Relative to this crude calculation, what are some factors you would want to constrain better from observations if you might want to improve estimates of the contribution of land ice to sea level rise (give two examples, with your reasoning)

Explanation / Answer

a) Energy needed to melt ice of 1.5km (1500m),

E = Thickness * density * latent heat = 1500 * 0.9 * 103 * 3.3 * 105 = 4.4* 1011 J/m2 or 4.4* 1011 W-s/m2

Therefore, the time required to melt the entire 1.5km thick ice sheet is,

t = energy needed / energy imbalance = 4.4* 1011 W-s/m2 / 9 W/m2 = 4.95 * 1010 s

Converting it into years, t = 4.95 * 1010 / (86400 * 365) = 1569.6 Years = 1.6 ky

Dividing thi time by the total thickness will give us the melt per year,

Melt per year = 1500 / 1569.6 = 0.95 m/year

b)  Melt per year = 0.95 m/year

Melt per decade = 0.95 * 10 = 9.5 m/decade

Area of greenland affected, A = 0.2 * (1.7*106) = 3.4 * 105 km2 = 3.4 * 1011 m2

Therefore, the volume of melted ice,

V = 9.5 * 3.4 * 1011 = 3.23 * 1012 m3/decade

This is the volume of ice melted. But remember that the ice and water have different densities and therefore, the melted ice volume is not equal to the volume of water generated. To calculate this, the weight of ice melted must be calculated:

w = V*density = 3.23 * 1012 * 0.9 * 103 = 2.9 * 10 15 kg

Since the wieght of ice and water is always the same this is the weight of water added to the oceans.

We know the density of water is 1000 kg/m3

Therefore, volume of water added, V = w/density = 2.9 * 10 15 / 1000 = 2.9 * 10 12 m3

Since this volume of added water is assumed to be evenly distributed, the rise in water level will be,

rise = 2.9 * 1012 / 3.6 * 1014 m2 = 0.8 * 10-2 m/decade = 0.8 cm/decade

The two factors I would like to constrain better from observations would be the actual a) surface energy imbalance over time at Greenland, as this value is the base of our calculation of time and depth of melt and b) distribution of the melted ice that would contribute to sea level rise, as this rise will not be uniform.

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