Will the Ice Caps Melt?

"The engineer has learned vastly more from the steam-engine than the steam-engine will ever learn from the engineer."
   
-- Prof John B. Fenn, Nobel Prize, Chemistry, 2002

There is considerable debate over whether the "greenhouse gas" effect will raise the temperature of the atmosphere by between 1-5°C over the next 100 years. But even if you grant for the sake of argument the Warmist claim that the earth's atmosphere will go up a full five degrees Centigrade in temperature, Al Gore's claim that ocean levels will rise 20 feet thanks to global warming seems to ignore the laws of thermodynamics. I am no climatologist, but I do know about physics.

Anyone who has ever spent time in a temperate climate following a snowy winter realizes that when the air temperature rises above 32°F the snow and ice do not melt immediately. We may experience many balmy early spring days with temperatures well above freezing while snow drifts slowly melt over days or weeks. Similarly, lakes and ponds take some time to freeze even days or weeks after the air temperature has plunged below zero.  This is due to the latent heat of freezing/melting of water, a physical concept long quantified in thermodynamics.

That aspect of basic physics seems to have been overlooked by climatologists in their alarming claims of dramatic and rapid sea-level rise due to melting of the Antarctic ice caps and Greenland glaciers. But of course, we have learned that models predicting global warming also failed to take account of precipitation, so overlooking important factors ("inconvenient truths") should not cause much surprise anymore.

The scientific data necessary to calculate the amount of heat necessary to melt enough ice to raise ocean levels 20 feet is readily available on the internet, and the calculations needed to see if polar cap melting passes the laugh test are surprisingly simple. Nothing beyond multiplication and division, and because we will use metric measures for simplicity's sake, much of the multiplying is by ten or a factor of ten.

Let's review the math.  The logic and calculations are within the grasp of anyone who cares to focus on the subject for minute or two, and speak for themselves.

I should first mention that the only source of energy to heat the atmosphere is the sun.  The average energy per unit time (power) in the form of sunlight impinging on the earth is roughly constant year-to-year, and there are no means to increase or reduce the energy flux to the earth. The question merely is how much of this energy is trapped in the atmosphere and available to melt ice thus effecting "climate change".

How much heat must be trapped to raise the atmospheric temperature by a degree centigrade (or more)  can be readily calculated, knowing the mass of the atmosphere and the specific heat of air.  Specific heat is simply an empirically-determined quantity that corresponds to the number of units of heat energy required to raise a specific mass of a substance, in this case air, by 1 degree in temperature.  A common unit of energy familiar to most of us is the calorie.  But for simplicity, in this calculation I will use the MKS[*] metric unit of the Joule (J), which, while perhaps unfamiliar to many readers in itself, is the numerator in the definition of our common unit of power, the Watt[†] = Joule/second.

The mass of the atmosphere can be found here. We also know that it is principally composed of air, so without loss of accuracy in what is essentially an "order of magnitude" calculation, it is fair to employ the specific heat of air at constant pressure, Cp which also can be referenced on the internet here.  While this has a value that changes with temperature, it doesn't change by orders of magnitude, consequently, I choose the value at 0° C, which, as we all know, is near to the global mean temperature at sea level.  In this I err on the side of caution, overestimating the heat energy in the calculation below, because as we all know, both air pressure and temperature drop with altitude.  Also note that while the specific heat value cited uses the unit °K in the denominator, this is equal to a °C. I use the tilda (~) as symbol for "circa" or "approximately". 


Mass of atmosphere:

5 x 1018 kg

Specific heat of air:

1.005 kJ/kg-°C

Heat needed to raise the temp of the atmosphere 1° C:

~5 x 1018 kJ

Heat needed to raise the temp of the atmosphere 5° C:

~2.5 x 1019 kJ


It is instructive now to compare this quantity of heat with the amount that would be required to melt sufficient volume of ice from the Antarctic ice to raise the sea-level by 20-feet as predicted by Al Gore.  Although ice floats, ice and water are very close in density, so at first approximation, it is fair to say that the volume of sea-water required to raise sea-level by 20-feet would be equivalent to the volume of ice that would need to melt to fill the ocean basins in order to cause that rise.  Consequently, let's first roughly calculate the volume of seawater necessary. 

The surface area of the earth can be looked up here.  It is 5.1 x 108 square kilometers, which I convert to 5.1 x  1014 square meters below for the purpose of our calculation.  Al Gore's 20-foot-rise is equal to ~6 meter. Let's use the commonly cited figure that 70% of the earth's surface is covered by the oceans and seas.  Accordingly,


Area of earth's surface:

5.1 x 1014 m2

Proportion of earth's surface covered by water:

70%

Area of oceans and seas:

~3.6 x 1014 m2

Sea level rise predicted by Al Gore:

20 feet = 6 m

Volume of water necessary to raise sea-level 20-feet:

~22 x 1015 m3

Volume of ice that needs to melt to raise sea-level 20-feet:

~22 x 1015 m3


This is where the latent heat of melting comes into the equation.  As we all know, when we drop an ice cube into our glass of water, soft-drink or adult-beverage, it quickly cools the drink.  Heat is transferred to the ice from the liquid in order to melt the ice; this loss of heat cools and reduces the temperature of the liquid.  This cooling continues until the ice melts completely. 

Scientists have long known that a mixture if ice and water (ice-water) remains at the freezing / melting point (0° C = 32°F).  Adding heat does NOT change the temperature, it just melts more ice; withdrawing heat does NOT change the temperature it just freezes more water.  The temperature of ice-water will not rise until all the ice is melted; conversely, the temperature of ice-water will not fall until all the water is frozen.  The heat that would have otherwise raised the ice temperature is somehow "stored" in the melt water - hence "latent heat". 

As an aside, the transformation of the latent-heat of steam into work via steam-engines has had, and continues to have, vast industrial importance. The early systematic study of steam-engines in order to improve their performance, laid the groundwork for the science of thermodynamics, which undergirds essentially all of physics and chemistry.

It turns out that latent heats of melting (and evaporation) are generally very large quantities when compared to the amount of heat necessary to change temperatures.  Also, as usual in such analyses we normalize to units of mass. Since the density of water/ice is roughly a thousand times higher than air, this also greatly impacts the magnitudes of energy involved, as you will see below.  So let's proceed with the calculation. 

The latent heat of melting of water can be looked up here. It is 334 kJ/kg of water.  One of the benefits of the metric system is that 1 ml = 1 cm3 = 1 g of water; this "built in" conversion simplifies many engineering calculations.  Remembering this fact, we do not need to look up the density of water. Converting this density, 1g/cm3, to MKS units, yields density of water = 1000 kg/m3.  We now have all our data for the rough calculation:


Volume of ice that needs to melt (from above):

~22 x 1015

Density of water and ice:

1000 kg/m3

Mass of ice that needs to melt:

~22 x 1018 kg

Latent heat of melting for water

3.34 x 102 kJ/kg

Heat necessary to melt ice to achieve 20-foot sea-level rise

~ 7.4 x 1021 kJ


Following this "back of the envelope" calculation, let's compare the two energy values:


Heat needed to raise the temp of the atmosphere 5° C:

~2.5 x 1019 kJ

Heat necessary to melt ice to achieve 20-foot sea-level rise

~7.4    x 1021 kJ


"There is a difference of 30* between these two figures, by implication extending the time-horizon for sea-level rise from 100 to 3000 years at the earliest. This does NOT mean that ice caps have not melted in the distant past nor that ice-age glaciers have not grown to cover much of the northern hemisphere; it simply means that the time scales involved to move sufficient quantities of heat to effect such melting or freezing occur over what we scientists commonly call "geological" time scales, i.e. tens or hundreds of thousands of years.

Even if sufficient heat is trapped in the atmosphere to raise it the maximum value predicted by anthropogenic "global warming" alarmists (5°C) over the next 100 years, thirty  times more heat energy must be imparted into the ice-caps to melt sufficient ice to raise sea-levels the catastrophic levels prophesied by Al Gore."

*Editor's note: a transposed decimal point led to an incorrect multiple used here when this article was first published. The energy required is nevertheless hundreds of times greater than evidently assumed by Al Gore.

Jerome J. Schmitt has a degree in mechanical engineering from Yale, and is president of NanoEngineering Corporation.

[*] MKS = meter-kilogram-second instead of cgs units = centimeter-gram-second for the units of length, mass and time.

[†] -After James Watt, inventor of the first practical steam-engine which employed a separate condenser.
"The engineer has learned vastly more from the steam-engine than the steam-engine will ever learn from the engineer."
   
-- Prof John B. Fenn, Nobel Prize, Chemistry, 2002

There is considerable debate over whether the "greenhouse gas" effect will raise the temperature of the atmosphere by between 1-5°C over the next 100 years. But even if you grant for the sake of argument the Warmist claim that the earth's atmosphere will go up a full five degrees Centigrade in temperature, Al Gore's claim that ocean levels will rise 20 feet thanks to global warming seems to ignore the laws of thermodynamics. I am no climatologist, but I do know about physics.

Anyone who has ever spent time in a temperate climate following a snowy winter realizes that when the air temperature rises above 32°F the snow and ice do not melt immediately. We may experience many balmy early spring days with temperatures well above freezing while snow drifts slowly melt over days or weeks. Similarly, lakes and ponds take some time to freeze even days or weeks after the air temperature has plunged below zero.  This is due to the latent heat of freezing/melting of water, a physical concept long quantified in thermodynamics.

That aspect of basic physics seems to have been overlooked by climatologists in their alarming claims of dramatic and rapid sea-level rise due to melting of the Antarctic ice caps and Greenland glaciers. But of course, we have learned that models predicting global warming also failed to take account of precipitation, so overlooking important factors ("inconvenient truths") should not cause much surprise anymore.

The scientific data necessary to calculate the amount of heat necessary to melt enough ice to raise ocean levels 20 feet is readily available on the internet, and the calculations needed to see if polar cap melting passes the laugh test are surprisingly simple. Nothing beyond multiplication and division, and because we will use metric measures for simplicity's sake, much of the multiplying is by ten or a factor of ten.

Let's review the math.  The logic and calculations are within the grasp of anyone who cares to focus on the subject for minute or two, and speak for themselves.

I should first mention that the only source of energy to heat the atmosphere is the sun.  The average energy per unit time (power) in the form of sunlight impinging on the earth is roughly constant year-to-year, and there are no means to increase or reduce the energy flux to the earth. The question merely is how much of this energy is trapped in the atmosphere and available to melt ice thus effecting "climate change".

How much heat must be trapped to raise the atmospheric temperature by a degree centigrade (or more)  can be readily calculated, knowing the mass of the atmosphere and the specific heat of air.  Specific heat is simply an empirically-determined quantity that corresponds to the number of units of heat energy required to raise a specific mass of a substance, in this case air, by 1 degree in temperature.  A common unit of energy familiar to most of us is the calorie.  But for simplicity, in this calculation I will use the MKS[*] metric unit of the Joule (J), which, while perhaps unfamiliar to many readers in itself, is the numerator in the definition of our common unit of power, the Watt[†] = Joule/second.

The mass of the atmosphere can be found here. We also know that it is principally composed of air, so without loss of accuracy in what is essentially an "order of magnitude" calculation, it is fair to employ the specific heat of air at constant pressure, Cp which also can be referenced on the internet here.  While this has a value that changes with temperature, it doesn't change by orders of magnitude, consequently, I choose the value at 0° C, which, as we all know, is near to the global mean temperature at sea level.  In this I err on the side of caution, overestimating the heat energy in the calculation below, because as we all know, both air pressure and temperature drop with altitude.  Also note that while the specific heat value cited uses the unit °K in the denominator, this is equal to a °C. I use the tilda (~) as symbol for "circa" or "approximately". 


Mass of atmosphere:

5 x 1018 kg

Specific heat of air:

1.005 kJ/kg-°C

Heat needed to raise the temp of the atmosphere 1° C:

~5 x 1018 kJ

Heat needed to raise the temp of the atmosphere 5° C:

~2.5 x 1019 kJ


It is instructive now to compare this quantity of heat with the amount that would be required to melt sufficient volume of ice from the Antarctic ice to raise the sea-level by 20-feet as predicted by Al Gore.  Although ice floats, ice and water are very close in density, so at first approximation, it is fair to say that the volume of sea-water required to raise sea-level by 20-feet would be equivalent to the volume of ice that would need to melt to fill the ocean basins in order to cause that rise.  Consequently, let's first roughly calculate the volume of seawater necessary. 

The surface area of the earth can be looked up here.  It is 5.1 x 108 square kilometers, which I convert to 5.1 x  1014 square meters below for the purpose of our calculation.  Al Gore's 20-foot-rise is equal to ~6 meter. Let's use the commonly cited figure that 70% of the earth's surface is covered by the oceans and seas.  Accordingly,


Area of earth's surface:

5.1 x 1014 m2

Proportion of earth's surface covered by water:

70%

Area of oceans and seas:

~3.6 x 1014 m2

Sea level rise predicted by Al Gore:

20 feet = 6 m

Volume of water necessary to raise sea-level 20-feet:

~22 x 1015 m3

Volume of ice that needs to melt to raise sea-level 20-feet:

~22 x 1015 m3


This is where the latent heat of melting comes into the equation.  As we all know, when we drop an ice cube into our glass of water, soft-drink or adult-beverage, it quickly cools the drink.  Heat is transferred to the ice from the liquid in order to melt the ice; this loss of heat cools and reduces the temperature of the liquid.  This cooling continues until the ice melts completely. 

Scientists have long known that a mixture if ice and water (ice-water) remains at the freezing / melting point (0° C = 32°F).  Adding heat does NOT change the temperature, it just melts more ice; withdrawing heat does NOT change the temperature it just freezes more water.  The temperature of ice-water will not rise until all the ice is melted; conversely, the temperature of ice-water will not fall until all the water is frozen.  The heat that would have otherwise raised the ice temperature is somehow "stored" in the melt water - hence "latent heat". 

As an aside, the transformation of the latent-heat of steam into work via steam-engines has had, and continues to have, vast industrial importance. The early systematic study of steam-engines in order to improve their performance, laid the groundwork for the science of thermodynamics, which undergirds essentially all of physics and chemistry.

It turns out that latent heats of melting (and evaporation) are generally very large quantities when compared to the amount of heat necessary to change temperatures.  Also, as usual in such analyses we normalize to units of mass. Since the density of water/ice is roughly a thousand times higher than air, this also greatly impacts the magnitudes of energy involved, as you will see below.  So let's proceed with the calculation. 

The latent heat of melting of water can be looked up here. It is 334 kJ/kg of water.  One of the benefits of the metric system is that 1 ml = 1 cm3 = 1 g of water; this "built in" conversion simplifies many engineering calculations.  Remembering this fact, we do not need to look up the density of water. Converting this density, 1g/cm3, to MKS units, yields density of water = 1000 kg/m3.  We now have all our data for the rough calculation:


Volume of ice that needs to melt (from above):

~22 x 1015

Density of water and ice:

1000 kg/m3

Mass of ice that needs to melt:

~22 x 1018 kg

Latent heat of melting for water

3.34 x 102 kJ/kg

Heat necessary to melt ice to achieve 20-foot sea-level rise

~ 7.4 x 1021 kJ


Following this "back of the envelope" calculation, let's compare the two energy values:


Heat needed to raise the temp of the atmosphere 5° C:

~2.5 x 1019 kJ

Heat necessary to melt ice to achieve 20-foot sea-level rise

~7.4    x 1021 kJ


"There is a difference of 30* between these two figures, by implication extending the time-horizon for sea-level rise from 100 to 3000 years at the earliest. This does NOT mean that ice caps have not melted in the distant past nor that ice-age glaciers have not grown to cover much of the northern hemisphere; it simply means that the time scales involved to move sufficient quantities of heat to effect such melting or freezing occur over what we scientists commonly call "geological" time scales, i.e. tens or hundreds of thousands of years.

Even if sufficient heat is trapped in the atmosphere to raise it the maximum value predicted by anthropogenic "global warming" alarmists (5°C) over the next 100 years, thirty  times more heat energy must be imparted into the ice-caps to melt sufficient ice to raise sea-levels the catastrophic levels prophesied by Al Gore."

*Editor's note: a transposed decimal point led to an incorrect multiple used here when this article was first published. The energy required is nevertheless hundreds of times greater than evidently assumed by Al Gore.

Jerome J. Schmitt has a degree in mechanical engineering from Yale, and is president of NanoEngineering Corporation.

[*] MKS = meter-kilogram-second instead of cgs units = centimeter-gram-second for the units of length, mass and time.

[†] -After James Watt, inventor of the first practical steam-engine which employed a separate condenser.