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SEA LEVEL

By Charles Rhodes, P.Eng., Ph.D.

PEAK SEA LEVEL:
One of the major consequences of global warming is a long term increase in world wide average sea level. However, it is the increase in local peak sea level that causes flood and storm damage. The local peak sea level has multiple components. There is the average sea level that is increased by melting of land borne ice. There is the daily tidal change that is caused by rotation of the Earth about its axis in the proximity of the moon and the sun. There is a nearly monthly tidal change caused by orbit of the Earth-Moon system about its common center of mass in the proximity of the sun. There are local sea level changes caused by changes in atmospheric pressure and wind. Finally there are sea level changes caused by sea floor geometries that convert horizontal liquid kinetic energy into vertical gravitational potential energy of position. Dangerous conditions and storm damage normally occur when these various effects combine to produce a temporary local sea level that is much higher than normal.

Global warming affects the average sea level via melting of land borne ice and thermal expansion of the ocean. Global warming affects the average atmospheric water vapor content via sea surface warming, which in turn affects both the emissivity. Quantification of the increase in local peak sea level due to global warming triggered storm activity is beyond the scope of this analysis. The amount of damage caused by the sea at a particular location is determined by the coincidence of a very low atmospheric pressure due to a storm with a high on-shore wind and a very high tide due to an unfavourable Earth-Moon-Sun alignment.
 

STRATIFIED OCEAN:
In the tropical ocean, where the ocean surface temperature is greater than 4 degrees C, lower density warm water floats on top of higher density cool water. Addition of heat to the ocean surface increases this temperature stratification causing the ocean surface temperature to follow the average air temperature. Hence the net on-going heat absorption per unit area by the tropical ocean due to global warming is relatively small.
 

MIXED OCEAN:
In the polar ocean, where the ocean surface temperature is less than 4 degrees C, lower density cold water floats on top of higher density warmer water. Addition of heat to the ocean surface increases the density of the surface water causing the surface water sink. Hence the polar ocean is relatively well mixed and the ocean surface temperature remains almost constant. This surface temperature is further fixed by the presence of floating ice, which absorbs heat at a constant temperature by melting. Thus the polar ocean surface temperature does not follow the air temperature. Hence the net on-going heat absorption per unit area by the open portion of the polar ocean due to global warming is relatively large.
 

TIME CONSTANTS:
The rate of sea level rise is subject to three time constants.
1. The exponential decay time constant of transient CO2 in the atmosphere, This time constant of about 41 years is a reflection of the lifetime of transient CO2 injected into the Earth's atmosphere.

2. The natural time delay due to formation or melting of floating polar ice. Under natural drivers this time constant is several thousand years but under 20 fold higher CO2 injection by combustion of fossil fuels this time constant decreases to about a half century.

3. The delay time between a step change in ocean surface temperature and the corresponding change in average ocean temperature. Due to ocean thermal stratification this time constant is many thousands of years.
 

RATE OF RISE OF AVERAGE SEA LEVEL DUE TO MELTING OF LAND BORNE ICE:
Assume that the rate of rise of average sea level is dominated by the rate at which net heat absorbed by the ocean melts land borne ice on Greenland and Antarctica. This rate has a firm upper limit set by the maximum rate at which net heat can be absorbed by the oceans.

From the web page: Thermal Runaway At steady state prior to the industrial revolution the maximum solar power available for absorption by the ocean is:
240.15 W / m^2. If the thermal emission temperature failed to follow changes in emissivity up to the threshold of thermal runaway the average heat absorpion would increase by a factor:
(Tac / Taa)^4 = (1.013355593)^4
= 1.054502164
so the maximum possible average net heat absorption by the ocean would be:
0.054502164 (240.15 W / m^2) = 13.08869 W / m^2

Once thermal runaway comes into play the maximum possible average heat absorption by the ocean would be given by :
(Tae / Taa)^4 = (1.047755266)^4
= 1.205145293
so the maximum average heat absorption becomes:BR> 0.205145293 (240.15 W / m^2) = 49.2656 W / m^2.

Define:
Hf = heat of fusion of water = 79.72 cal / gm
Then:
the amount of heat required to melt ice to form a cubic metre of water is given by:
79.72 cal / gm X 4.18 J / cal X 10^6 gm / m^3 = 333.2296 X 10^6 J / m^3

Then the maximum possible sea level rise rate under thermal runaway conditions would be:
[(49.2656 W / m^2) / (333.2296 X 10^6 J / m^3)] X (1 J / s-W) X (3600 S / hr X 8766 h / year)
= 4.66 m / year

Note that this extreme sea level rise rate assumes that all the floating polar ice has already melted.

In 1996 the equivalent heat absorption rate by the ocean was:
(Tab / Taa)^4 = (270.7 / 269.55)^4
= 1.017175002
giving a maximum net heat absorption rate of:
0.017175 (240.15 W / m^2) = 4.12457 W / m^2

Then if there was no floating ice the maximum possible sea level rise rate under 1996 conditions would have been:
[(4.12457 W / m^2) / (333.2296 X 10^6 J / m^3)] X (1 J / s-W) X (3600 S / hr X 8766 h / year)
= 0.3905 m / year

Note that in reality due to warm surface water floating on top of the bulk ocean the actual ocean surface temperature follows changes in Ft so that the net heat absorption rate is only about 1% of the maximum possible rate calculated above. The experimentally measured sea level rise rate is about 0.003 m / year, indicating that the thermal emission temperature closely follows the change in emissivity Ft.

If use of fossil fuels is ceased after the onset of this land borne ice melting process, the average sea level will continue rising until the excess atmospheric CO2 concentration decays. The exponential decay time constant To for excess CO2 in the atmosphere is about 41 years. Hence even if use of fossil fuels was stopped today the sea level would rise at least:
3 mm/ year X 41 years = 123 mm = 0.123 m

Note that if the use of fossil fuels is not stopped the oceans will keep on absorbing heat and the sea level will keep on rising until the supply of land borne ice that can melt or slip into the ocean is exhausted.
 

MELTING OF FLOATING POLAR ICE:
In 2007 the Alfred-Wegener-Institute for Polar and Marine Research expedition to the North Polar Sea found that large areas of the Arctic sea-ice were only 1 m thick, half the thickness found in 2001. The National Snow and Ice Data Center reported that in 2000 the minimum Arctic sea ice area was 6.74 X 10^6 km^2 and that in 2007 the corresponding minimum Arctic sea ice area was 4.13 X 10^6 km^2. Hence the average melting rate of Arctic sea ice over the period 2000 to 2007 is given by:
[(6.74 X 10^6 km^2 X .002 km) - (4.13 X 10^6 km^2 X .001 km)] / 7 years
= [13.48 X 10^3 km^3 - 4.13 X 10^3 km^3] / 7 years
= 1336 km^3 / year.

Satelite radar and gravity measurements of the Greenland glacier melting rate conducted during the period 2002 to 2005 indicate that this glacier's melting rate was about 230 km^3 / year.

Hence the total Arctic ice melting rate in during the period 2002 to 2005 was about:
1336 km^3 / year + 230 km^3 / year = 1566 km^3 / year.

The increase in average sea level due to melting of land borne ice is given by:
(230 km^3 / year) / Ao = [(230 km^3 / year) / (361 X 10^12 m^2)] X (10^9 m^3 / km^3)
= .000637 m/year = 0.637 mm/year
which is consistent with the experimentally measured annual increase in average sea level of about 3 mm / year.
 

AVERAGE OCEAN SURFACE TEMPERATURE:
Over thousands of years prior to the industrial revolution the ocean acquired its present temperature profile. The average surface temperature is similar to the average surface temperature for an ideal rotating sphere modified by the Greenhouse Effect. It is shown in Surface Temperature of an Ideal Rotating Body that the average surface temperature Tea of an ideal Earth is:
Tea = 278.636 degrees K.

It is shown in the section titled Temperature Data that the historical Greenhouse Effect produces a warming of 8.36 degrees K.

Hence the calculated average ocean surface temperature Taa is given by:
Taa = 278.636 K + 8.36 K
= 286.996 degrees K
= 286.996 - 273.15
= 13.846 degrees C
This value compares with an average ground level Earth surface temperature of about 15 C.
 

THERMAL EXPANSION:
In the tropics ocean water stratifies, so the surface temperature is not representative of the bulk ocean temperature. Ocean water is most dense at about 4 degrees C. Most of the bulk ocean below the surface layer is in the temperature range -2 to 5 degrees C. Since at 4 degrees C the temperature coefficient of expansion is zero, the net change in sea level caused by global warming induced thermal expansion is uncertain.

OCEAN RESPONSE TIME PERIOD:
On the web page titled Radiation Physics it was assumed that the time intervals involved were much smaller than the time required for the ocean to come to steady state. To justify this assumption the response time Tv of the mixed ocean is found below.

Assume that over sufficient time ocean currents will distribute heat from the mixed ocean, where the heat is captured, to the stratified ocean where there is little ongoing heat capture.

Let Tao = average bulk ocean temperature
Let Tas = average ocean surface temperature
Assume that to restore steady state condions:
dTao = dTas = dTa

Conservation of energy gives:
Cp Mo dTao = Ht X Tv
or
Tv = (Cp Mo dTao) / (Ht) where:
Cp = the heat capacity of water = 1 cal / gm-deg C
Mo = mass of oceans
Ht = rate of heat capture

On this web page it was previously found that for a mixed ocean of area Ap the total heat flux Ht captured is:
Ht = Ap (dTa / Ta) Ho (1 - Fr)
Thus:
Tv = Cp Mo dTao / [Ap (dTa / Ta) Ho (1 - Fr)]
= Cp Mo Ta / [ Ap Ho (1 - Fr)]
Let Do = average depth of oceans = 3790 m
Let Ro = average density of oceans ~ 1 gm / cm^3
Recall that:
Ao = area of oceans,
so:
Mo = Ao Do Ro
Hence:
Tv = Cp Ao Do Ro Ta / (Ap Ho (1 - Fr))
= [Cp Do Ro Ta / (Ho (1 - Fr))] [Ao / Ap]
= [(1 cal / gm-deg K)(3790 m)(1 gm / cm^3)(270 deg K)(10^6 cm^3 / m^3)(4.18 J / cal)(1 W-s /J)]
/ [(1367 W / m^2) (1 - .297)][Ao / Ap]
= [3790 X 270 X 10^6 X 4.18 s] / [1367 X .703][Ao / Ap]
= 4451 X 10^6 s [Ao / Ap]
= (4451 X 10^6 s) [Ao / Ap] X (1h / 3600 s) X (1 year / 8766 h)
= 141.04 years [Ao / Ap]
Thus the response time Tv of the ocean to a step change in Ft is about:
Tv = 141.04 years [Ao / Ap]
Recall that Ap is the area of mixed ocean which has a surface temperature less than 4 degrees C. To a first approximation we can take Ap as the area of the Arctic Ocean, which gives a value of the bulk ocean time constant of:
Tv = 141.04 years X [(361 X 10^12 m^2) / (14.056 X 10^12 m^2)]
= 3622 years
Note that due to the uncertainty in the actual value of Ap, there could easily be a 1000 year error in this calculated result. However, Tv is long enough that it may be significant in determination of interglacial periods.
 

THE SEA LEVEL CRUNCH:
One of the serious long term consequences of an increase in atmospheric carbon dioxide concentration is an increase in sea level. Many of the world's human population concentrations are located at sea ports and on river deltas. These population concentrations are seriously threatened by an increase in sea level of only a few metres.

1. Historical data over the period 1910 to 1990 indicates an average sea level increase over that period of about 1.9 mm / year. During the period 1990 to 2009 this rate of sea level increase approximately doubled.
2. The sedimentary record shows that during an interglacial period about 125,000 years ago for a short time the average sea level reached 4 m to 6 m higher than its present level.
3. The volume of ice on Greenland, if it all melted, would increase the average sea level by 7.2 m.
4. The volume of ice on Antarctica, if it all melted, would increase the average sea level by 61.1 m.
5. The volume of ice in the grounded interior reservoir of the West Antarctic Ice Sheet, if it all melted, would raise the average sea level by 5 to 6 m.
6. If there is little floating ice in the oceans except around Greenland and Antactica, where the floating sea ice is maintained by glacier bottom discharge, and if there is sufficient natural ocean circulation that average sea level rise is determined by ocean net heat absorption causing melting of this floating sea ice, then the projected average sea level rise rate when the atmospheric carbon dioxide concentration reaches twice its historic value, is about .0384 m / year.
7. The average sea level will continue rising until the excess atmospheric CO2 concentration decays, or until the average ocean temperature responds or until the ice caps melt. Once the ice cap shrinking process starts, even if the injection of fossil CO2 is stopped, the melting process will continue due to the excess atmospheric CO2 concentration exponential decay time constant To, which is about 41 years.
8. When the above conditions are considered together with the ice core record of atmospheric CO2 concentration, it appears that an eventual rise in average sea level of at least:
1.38 m to 2.76 m
will occur. To the extent that there is a delay in controlling the rate of release of fossil CO2 to the atmosphere this rise in average sea level will increase. The initiating factor will be loss of perimeter floating ice around Greenland and Antarctica, allowing high pressure fluid under the land borne ice caps to flow into the surrounding ocean.

Thus, a 2 m increase in average sea level could easily occur by the year 2100. Present Canadian government plans do not contemplate serious CO2 emission reductions before the year 2050, and the value of the excess CO2 exponential decay time constant To, relating to the decay of excess atmospheric CO2, would then extend the excess CO2 heating out to past the year 2100. It will take a major change in government priorities to significantly alter this course of events.

GLOSSARY OF TERMS

This web page last updated October 13, 2014.

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