15. The truth about sea level rise

One of the most emotive and alarmist claims made by climate scientists is that global warming will lead to a catastrophic sea level rise (SLR) that will submerge major cities and lead to an unprecedented humanitarian crisis and global extinction event.

Fig. 15.1: Britain's favourite polar bear - Peppy.

On the face of it this seems quite plausible, even likely. We see images of collapsing ice shelves and retreating glaciers on an almost daily basis. We see icebergs the size of cities being calved off from Antarctica and then polar bears looking forlorn on icebergs the size of a lifeboat, like something from a Fox’s glacier mint advert. So what is the reality?

Fig. 15.2: Life imitating art.

There are two principal ways that sea levels might change: either the amount of water in the sea changes, or the existing sea water changes its density. The former can happen if ice caps on Greenland or Antarctica melt. It cannot happen through the melting of sea ice because of Archimedes' Principle as I explained in Post 2, nor for the same reason can an increase in sea ice change the sea level. One alternatively mechanism is through increased evaporation of condensation, but that requires the humidity of the atmosphere to change. As for density changes, these are governed mainly by changes in the temperature of the sea water.


Scenario 1: Thermal Expansion

In the case of rising sea temperatures it is not the current global temperature that is important per se, but the temperature history and the thermal budget of the Earth. Global temperature rises can only raise sea levels directly (excluding from ice melting) by thermal expansion of the sea water. As I pointed out in Post 2, the coefficient of thermal expansion by volume for water is 0.000207 per degree Celsius or 207 ppm/°C. So a column of water 1000 metres or 1 km high will increase in height by only 20.7 cm if its temperature increases by 1 °C (or 1 K where K is the unit of absolute thermodynamic temperature - the kelvin).

But there is another factor we need to consider - the total heat or thermal energy required to do this. This is because this energy needs to come from somewhere, and once used it remains trapped in the water. It is, therefore, energy that has been sequestrated, the effect of which is to create an imbalance between the amount of energy the Earth receives from the Sun, and the amount it emits back out into space. This difference can only come from the net energy imbalance of the Earth’s energy budget.

In the last post we saw that this imbalance is currently estimated to be as much as (and no-one is saying it is more than) 0.9 W/m². If this figure is true (and as I pointed out, there is enormous uncertainty over its accuracy), and if it had been constant over the last 100 years (which is very unlikely), it would imply that the Earth has absorbed a total of 4.591x10¹⁴ joules of energy every second in that time period, or 1.45 x 10²⁴ J in total. Of course that is the upper limit of what is likely. No-one seriously thinks the Earth’s energy imbalance has always been 0.9 W/m². So the average over the last 100 years must be considerably less, and probably less than half.

Whatever the value, though, this heat will increase the temperature of the oceans. The question is, by how much, or to what depth?

As the specific heat capacity of water is 4200 Jkg^-1K^-1, the amount of energy required to heat 1 kg of water by 1 K (or 1 °C as these temperature changes are the same) will be 4200 J. If our 1 km high water column has a cross-section of 1 m², then it will contain 1000 tonnes of waters. Therefore, the total energy required to raise the temperature of the entire column by 1 °C will be 1,000,000 times greater than 4200 J, in other words 4.2 x 10⁹ J. As 70.8% of the Earth’s surface is covered by the oceans, the total volume of water down to a depth of 1 km will be 3.61 x 10¹⁷ m³. The total mass will be 3.61 x 10²⁰ kg, and the total heat capacity will be 4200 times higher still at 1.52 x 10²⁴ J/°C. So the mean temperature rise of the oceans down to a depth of 1 km over then last 100 years will be (at the absolute maximum) 1.45 x 10²⁴ ÷ (1.52 x 10²⁴) = 0.95 °C.

So, if we assume that the Earth’s energy imbalance over the last 100 years has been 0.9 W/m² everywhere and at all times, and if we assume that this heat has all ended up in the ocean, and it has heated the top 1000 m only, then the temperature rise of that top layer of water will be 0.95 °C. Some may find this number suspiciously close to the value claimed for global warming of about 1 °C per century. In other words, that climate scientists have worked backwards. They have assumed that the oceans must heat up by the same amount as the land over the same period, and to a depth of up to 1000 m, and worked out the amount of heat required to do this. From this they have inferred an imbalance in the Earth’s thermal budget rather than measured it.

Whatever the sequence of events, the resulting sea level rise (SLR) will be 197 mm (i.e. 0.95 x 207). If the warming layer of the ocean is thinner (say 500 m) but the surface energy imbalance is still 0.9 W/m², its mean temperature rise will be greater (an unlikely 1.90 °C for a 500 m thick layer), but the SLR will be the same, in other words a massive 1.97 mm per year. So what is clear is that the maximum sea level rise that can occur depends on the energy imbalance and not the water depth. In reality, the surface energy imbalance may be less than 0.9 W/m² (G. L. Stephens et al., Nature Geoscience 5, 691–696 (2012) suggest 0.6 W/m² as I pointed out in Post 13) and has in all likelihood got worse over time. So while it may be 0.9 W/m² now, it has probably averaged less than half of 0.9 W/m² over the last 100 years as global temperatures have risen. In which case the SLR has probably been less than 100 mm over the last century (or less than 66 mm if Stephens et al. are correct). But what about the future?

Assuming that the surface energy imbalance remains at 0.9 W/m² for the foreseeable future, and if we now assume that only that portion of the surface energy imbalance over the oceans is actually absorbed by the oceans, then the heat absorbed by the oceans each year will be 28.4 MJ/m². If this were absorbed by a column of water 1000 m deep it would result in a temperature rise of 6.76 mK. If it were instead absorbed by a column of water only 100 m deep it would result in a temperature rise of 67.6 mK. Either way, the resulting thermal expansion would be 1.40 mm per year. Even over 100 years this is a long way short of the 10 m rise some doom-mongers are projecting, and on its own is unlikely to pose a major threat to human civilization or the planet.

But thermal expansion is just one component of the overall problem, because not all the 0.9 W/m² (or 0.6 W/m²) need end up in the oceans. Some may instead end up melting the ice caps.


Scenario 2: Melting Ice Caps

In this scenario there are a number of factors that we need to identify and address. Firstly, where is the ice? If it is floating on the ocean surface then it cannot add to the sea level increase when it melts, despite 9% of the ice being above the water line. This is because of Archimedes’ principle as I explained in Post 2. It can, though, sequestrate energy and actually reduce warming elsewhere. The only ice that can increase sea levels when it melts is ice that is on land. This is found mainly on Greenland (3 million cubic kilometres) and Antarctica (30 million cubic kilometres).

Then there is the question of the ice temperature. Before it can melt it needs to be heated to 0 °C, yet the mean temperature of the ice on Antarctica is about -50 °C. In order to raise the ice temperature to melting point would require 3.20 x 10²⁴ J of energy (the specific heat capacity of ice is 2108 Jkg^-1K^-1 and its relative density is 0.92).

Next, you need to melt the ice. This will require an energy input of 1.01 x 10²⁵ J (the specific latent heat of ice is 332 kJ/kg); and then you need to heat it to the ambient temperature of the Earth, about 15 °C, otherwise it will cool the oceans and your global temperatures will go down. This requires another 1.91 x 10²⁴ J of energy. So the total energy required is 1.52 x 10²⁵ J. Given that the amount of power available to achieve this is at most only 0.9 W/m², that means it would take at least 1100 years to occur. But even that assumes that all the power from the Earth’s energy imbalance could be channelled somehow into melting the ice caps and nothing else.

In reality most will initially go into the oceans. As Antarctica and Greenland comprise only 3% of the Earth’s surface area, a more realistic estimate is that it would take up to 30,000 years, by which time we would be in the next ice age. Recent studies of Greenland appear to confirm this as they show Greenland has lost less that 0.05% of its ice in the last 10 years, but this may just be cyclical. 

One final point of note: while the total volume of ice on Antarctica is 30 million cubic kilometres, almost a third of this is below sea level. The net result is that if all the ice on Antarctica and Greenland were to melt, sea levels would rise by 65 metres not 91 metres. Yet over 30,000 years this will amount to a rise of only about 2 mm per year.


Scenario 3: Evaporation

The final possibility is that the sea water might just evaporate into the atmosphere leading to a loss of volume in the sea rather than a gain. Approximately 0.4% of the atmosphere by volume is water vapour. But if all this water were to suddenly condense out of the atmosphere it would only add 3.6 cm to the depth of the oceans. Given that the ambient temperature at the surface of the Earth is 15 °C, while the humidity at 0 °C would be expected to drop to near zero, this suggests that an increase of 1 °C in the surface temperature of the air would lead to a decrease in sea levels of around 2.4 mm. That is a pretty crude estimate, though, and assumes that the vapour pressure of water increases proportionately with temperature from its freezing point. A more accurate estimate can be made using the Clausius-Clapeyron equation.

Fig. 15.3: Schematic of phase boundaries on a P-T diagram.

This equation (see Eq. 15.1) relates the the slope of a phase boundary in a pressure-temperature diagram to the thermodynamic temperature, T, the molar latent heat for the phase change, L, and the change in molar volume across the boundary ∆V. An example of a phase diagram is shown in Fig. 15.3 above.

(15.1)

For a change from liquid to gas (as in evaporation) the term ∆V should be the difference in molar volumes between the water in the liquid phase and the vapour in the gas phase. However, as the volume of the vapour at the relevant pressures we are likely to encounter (i.e. around atmospheric pressure) is so much greater than it is for water (in fact by more than a factor of 1000) we can use the ideal gas law in Eq. 15.2 to substitute the molar volume of water vapour V for ∆V on the basis that the molar volume of the liquid is negligible.

(15.2)

In Eq. 15.2 the term V is the volume of one mole of water vapour at a pressure P and a temperature T. The term R is the molar gas constant where R = 8.314 Jmol^-1K^-1 and n is the molar density in mol/m³. The approximation of V for ∆V allows us to make a substitution from Eq. 15.2 into Eq. 15.1 to generate Eq. 15.3 which is now a function of only two variables, P and T.

(15.3)

This allows us to relate fractional changes in the pressure of the gaseous phase to fractional changes in temperature along the phase boundary. The differential in Eq. 15.3 implies that for small changes in P and T the following relation holds

(15.4)

while Eq. 15.2 yields the following relation between P, T and n.

(15.5)

Equating Eq. 15.4 with Eq. 15.5 gives the result for the fractional change in molar concentration of the vapour that will occur due to  evaporation across the phase boundary for a temperature change ∆T.

(15.6)

The relation in Eq. 15.6 allows us to estimate the change in water vapour concentration n as the temperature T changes. So for example, if the temperature T = 288 K and the latent heat of evaporation of water is 40.8 kJmol^-1K^-1, then a temperature rise of ∆T = 1 K will yield a fractional change of water vapour concentration of 0.0557. That in turn implies a total fall in sea level over the period of the temperature rise (which is about 100 years) of 2.0 mm (= 0.0557x36). Reassuringly, this is not that dissimilar to our original estimate of 2.4 mm, thus demonstrating two important points. Firstly, that the result is robust and consistent. Secondly, that our original back-of-the-envelope approximation, much loved by physicists everywhere, did not let us down.

The advantage of calculating the fractional change in n using Eq. 15.6 rather than calculating ∆n directly is that it means that we can avoid the complication of working out the relative humidity. The Clausius-Clapeyron equation, strictly speaking, only applies to closed systems in equilibrium, i.e. at 100% relative humidity. In open systems, such as the Earth's atmosphere above large oceans, the humidity is always less than the maximum. But Eq. 15.6 effectively removes the issue of relative humidity as it just introduces an additional scaling term that applies more or less equally to n and ∆n. Therefore it cancels out in Eq. 15.6. All of this may be somewhat pedantic, however, as the sea level fall due to evaporation is a full two orders of magnitude less than the previous two effects considered.


So the conclusion is this: prophesies of apocalyptic rising sea levels and submerging cities are still just alarmist nonsense. The physics proves that there is currently not enough energy available to achieve this on the timescale that some climate scientists predict, at least not yet. Thermal expansion and melting ice caps will each add not much more than 2 mm per year to sea levels. A recent paper by Anny Cazenave et al. (Advances in Space Research 62(7) 1639-1653 (2018) ) puts the sea level rise (SLR) from all sources at about 3.5 mm per year for the period 2005-2015  (see Fig. 15.4 below). These numbers do appear to be more consistent with the surface energy imbalance of 0.9 W/m² reported by Trenberth and co-workers rather than the 0.6 W/m² of Stephens et al., particularly the thermal expansion component.

Fig. 15.4: Possible breakdown of different contributions to sea level rise (1993-2015) from Cazenave et al.

One of the striking features of Fig. 15.4 in my view is the low contribution to sea level rise from ice melt in Antarctica compared to that from glaciers. Three explanations spring to mind. Firstly, there is probably more warming in the Northern Hemisphere because that is where the heat is being generated. Secondly, the glaciers in Europe are very close to the source of that heating. And thirdly, the ice in Antarctica is much colder than that in alpine glaciers, and so requires more heat to melt it. So, as I pointed out in the last post, this could mean that alpine glaciers will continue to recede, not because of CO₂ emissions, but because of local human industrial activity that leads to surface heating of the local environment, and thus a temperature rise of more than 0.3 °C above pre-industrial levels.