A Simple Estimate of Global Mean Sea Level Rise due to Increase in Global Temperatures

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During the 20th century, mean sea-level rose by about 10 cm and is expected to rise at a much faster rate in future. Rising sea-level increases the intensity of severe stroms and  tides causing serious flooding.  Cities and infrastructure near coastlines are vulnerable to damage. Rising sea-level also increases the risk of coastal erosion and shoreline retreat. 

Climate change is given as the reason for the accelerated rise in global mean sea-level and under current scenarios, we should expect a further rise of 60 to 100 cm by the year 2100.  For a lay person, it is difficult to relate a warming climate to a sea-level rise, and how the estimates are made might appear confusing.  

In this blog, I shall calculate sea-level rise due to melting of ice on land and thermal expansion of water. Both estimates may be reliably made using school level arithmatic.  I shall use  numbers rounded to two decimal places to make the calculations look tidy.

Thermal Expansion of water:  Water expands with rising temperature. For each degree centigrade rise in temperature, the volume of water increases by a factor equal to 2.14 x 10-4.   The average sea surface temperature is 17oC and during the 20th century the surface water temperature  increased by ~ 0.5oC. 

The interesting part of this calculation is to appreciate that the water temperature in the sea drops steadily with depth and reaches 4oC
 at about 700 m.  At 4oC, water has the highest density and sea water below 700 m does not change much in temperature - hence does not expand in volume and does not contribute to sea-level rise. 

When the sea warms, it is mainly the layers up to 700 m depth that warm and expand.  We can calculate the increase v in the volume V of water  when the temperature changes by dT.  Since the surface area A of the sea may be assumed constant, the increase h for dT = 0.5oC
 is  
                
                  v = 2.14 x 10-4 x 0.5 x V 
or               h A = 2.14 x 10-4 x 0.5 x 700 A
or               h = 0.075 metres = 7.5 cm in 100 years 

Over the past 20 years, the global warming rate has accelerated and it is estmated (IPCC 2013 Report) that in the next 100 years the sea level will rise by at least 20 cm due to thermal expansion of water (IPCC Senario RCP4.5).

Melting of Ice on Land:  At present, glaciers cover 10% of the land area and store about 75% of world's fresh water (of all the water on Earth, only about 2.5% is fresh water).  Twenty thousand years ago, at the peak of the last ice age, glaciers covered ~32% of the total land area!  
Ice is also present in the sea in the Arctic Ocean; but the  melting of ocean ice, as in the Arctic Ocean, does not change the sea-level.  It is only the land ice that can add additional water to the seas and cause its level to rise.
Ice on land is present in inland glaciers and ice sheets - the Greenland Ice Sheet (GIS) and Antarctic Ice Sheet (AIS). So far, the inland glaciers have contributed the largest amount of  melt-water; IPCC estimate (5th asssessment report 2013, chapter 13) that melting inland glaciers will contribute 12 ± 6 cm sea-level rise by the year 2100. 
  

The Greenland Ice Sheet, GIS covers 80% of Greenland surface and has an area equal to 1.7 million km2, its thickness on average is 2 km.  GIS is losing ice quite rapidly; during the decade 2002 to 2011, the average ice loss was 215 billion tons per year!

Antarctic Ice Sheet, AIS is much bigger.  It covers 98% of Antarctic continent and has an area 14 million km2, and is up to 4 km thick at some places.  It contains 26 million billion tons of ice.  While most parts of the ice sheet are considered reasonably stable, over the past few years, loss of ice from West AIS has been increasing and is currently about 80 billion tons per year.


Before we calculate the rise in sea level due to melting of the ice-sheets, let us note that the Earth has a surface area of 510 million km2. Of this about 71% is ocean - 360 million km2 and the land is 150 million km2.



GIS is losing 215 billion tons or 215 x 1012 kg of ice per year (1 ton is equal to 103 kg).  The density of ice is 917 kg m-3. Therefore, volume of water produced per year =  215 x 1012/917 or 2.35 x 1011m3.  
This water will raise the level of sea by 2.35 x1011m3/ 360 x 1012m2  = 0.653 mm per year
In 100 years GIS will increase the sea level by 6.5 cm.
However, it is expected that the rate of ice loss will increase significantly due to continued warming of the planet resulting in a much greater rise of global mean sea level due to GIS ice loss. 

The amount of ice lost per year by the Antarctic Ice Sheet is about 80 billion tons per year and will contribute only 2 cm to the rise of sea level over the next 100 years. This might be a gross underestimate of what could happen if our planet continues to heat up.  AIS has large areas of ice plates (shelves) projecting over the sea with their base in contact with the sea water (the grounding line). Warmer sea water is already melting the underside of these plates and adding extra water to the oceans. See slide.

The grounding line also moves inland increasing  the probability of thinned ice shelves breaking and falling in the ocean to raise sea levels. IPCC 2013 report estimates that, by the year 2100,  this (Ice Sheet Rapid Dynamics) can add 10 cm to the sea level rise. 

But what happens if all of the GIS and AIS melt over the next few hundred years.  How much the sea level rise be?

Considering that GIS is on average 2 km thick and covers an area 80% of 1.71 million km2; the volume of water that the ice in GIS will generate is 2.5 million km3; The resulting sea level rise will be 7 metres.  At 250 billion tons of ice lost per year, it will take GIS 10000 years to melt completely; however, the melting rates are expected to increase and many scientists think that GIS might melt away within the next 1000 years. The 7 meter sea-level rise will be catastrophic for our civilisation. 

See Also: https://www.bbc.co.uk/news/science-environment-48337629 
              May 21, 2019 report
See also https://www.nationalgeographic.com/environment/global-warming/sea-level-rise/
                           February 2019
Interesting Read:  https://edition.cnn.com/interactive/2022/04/world/climate-sea-level-rise-iceland-marshall-islands-cmd-intl/ 

Myths and Hypes about the Ubiquity of the Amazing Golden Ratio; Its Relation to Fibonacci Numbers; Logarithmic Spiral, Phyllotaxis and the Pentagram

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'In nature, Golden Ratio (Phi) and Fibonacci Numbers (FN)  are common, probably reflecting the practicalities of life.  In our human world, we might see them where they don't exist, but where they do, we find them pleasing.  Whether we are attracted to them by the mystic of mathematics or the aesthetics they produce is uncertain.'    ... Tim Entwise in Blueprint for Living  

Watch a pleasant 2.5 minute presentation on the Golden Ratio here but do not believe everything that is said in the video. 
Indeed, there is a lot of hype about Phi and FN -  the golden ratio, also known as the golden section or golden mean, is claimed even to connect humans to God!!  (It is claimed that in ɸ, He has crossed nothingness (0) by unity (1) to obtain the symbol for the golden ratio. But note that the symbol ɸ was adopted only recently for the golden ratio!).  It is stated that 'The Golden Section, or Phi, found throughout nature, also applies in undertanding the relationship of God to Creation'.


Many attempts have been made to bring order to the situation regarding hypes and myths about Phi and FN.  Please click here and here.  

Phi is an amazing number with some unique mathematical properties and this is what we shall be looking at in this blog - the aim is to give some examples invoving Phi and FN that have a wow factor - that is what recreational maths is all about.
First, we define the golden ratio and Fibonacci Numbers: The golden ratio is denoted by the greek letter capital Phi ɸ and its inverse by lower case phi φ.  
(Click on a slide to see full page image.  Press ESC to return to Text) 

The following slide is from

Fibonacci Numbers are intimately related to the Golden Ratio and are claimed to occur widely in nature.  I define them in the following slide - a more general and detailed discussion is available in my blog.

First, I would like to show how closely, Phi that is purely geometrical in origin is related to Fibonacci Numbers which are formed from numbers following a simple mathematical prescription.   

The second term in the expression for Fn becomes progressively smaller as n increases, and to a very good approximation Fn increases as ɸ to the power n for n greater than about 10.  
Also, notice that the second term is negative when n is even and positive for odd values of n. For small values of n, Fn oscillates about the values calculated for ɸ to the power n.  This is explained in the next slide:

Puzzle 1:  Climbing Steps:  The puzzle may be stated as follows:

You would like to climb six steps.  You can either climb one step at a time (s1) or two steps at a time (s2).  How many differnt ways can you climb the steps?

One way to solve the problem is to work sequencially 
First Step: - only one way --  1s1; N1 = 1
Second Step:  2s1 or 1s2.  N2 = 2
Third Step: 3s1; 1s1+1s2; 1s2+1s1.  N3 = 3
Fourth Step: 4s1; 2s1+1s2; 1s2+ 2s1; 1s1+1s2+1s1; 2s2. N4 = 5
Fifth Step: 5s1; 3s1+1s2; 2s1+1s2+1s1; 1s1+2s2; 1s1+1s2+2s1; 2s2+1s1; 1s2+1s1+1s2; 1s2+3s1.  N = 8  (6th Fibonacci number) 
Sixth Step:  6s1; 4s1+1s2; 3s1+1s2+1s1; 2s1+2s2; 2s1+1s2+2s1; 1s1+2s2+1s1; 1s1+1s2+3s1; 1s1+1s2+1s1+1s2; 3s2; 2s2+2s1; 1s2+4s1; 1s2+2s1+1s2; 1s2+1s1+1s2+1s1.  N = 13  (7th Fibinacci Number)

Notice the sequence of Fibonacci numbers 1, 2, 3, 5, 8, 13 appears here.  It is now straightforward to calculate the number of different ways you can climb 10 steps.  It is FN at n = 11 --- this is 144.  

Puzzle 2:  Seating Arrangement: 

At a school function with lots of children (C) and adults (A), the seating arrangement requires that an adult (A) must not sit next to another adult.  If there are N chairs then how many different ways they may be seated?

According to the puzzle,  combinations AA are not allowed.  Let us start with one chair

1 chair:   C or A.  N1 = 2
2 chairs: CC; CA; AC. N2 = 3
3 chairs: CCC; CCA; CAC; ACC; ACA. N3 = 5
4 chairs: CCCC; CCCA; CCAC; CACC, CACA; ACCC, ACCA; ACAC.  N4 = 8
5 chairs: CCCCC; CCCCA; CCCAC; CCACC; CCACA; CACCC; CACCA; CACAC; ACCCC; ACCCA; ACCAC; ACACC, ACACA.  N5 = 13
and so on...

Again the sequence of Fibonacci numbers 2, 3, 5, 8, 13,...appears. The number of different ways the visitors may be seated increases by ɸ = 1.618033 each time an extra chair is added.  


An Example from Biology:  Let us look at the family tree of bees. The situation may be stated as follows:
In a beehive, there is one female queen who lays all the eggs.  
If an egg is fertilized by a male bee, then the egg hatches into a female bee.  
But if the egg is not fertilized then it hatches into a male bee (a drone).  
Worker female bees do not lay eggs.

Essentially, a drone has one parent while a female bee has two parents. We wish to map out a family tree for bees.  I have prepared the next slide to show this:

Notice that at generation 8, the number of males, females and also anscestors increase by 1.6154 already.  The increase per generation will be equal to the Golden Ratio (=1.618) for n ≧ 10. 

Interestingly, FN and ɸ appear in the most unlikely places - before I discuss more examples, let us expand on the scope of the golden ratio that has  been defined for a line (one-dimension) so far.  Extend to two or three dimensions and we encounter some fascinating observations.

Golden Rectangle:   The sides of a golden rectangle are in the ratio ɸ = 1.618033.  

Claims have been made that the proportions (aspect ratio) of a golden rectangle are aesthetically most pleasing, and this is reflected in architectural designs; art; paintings; aspect ratio of books, cards and many other objects.  I think it is fair to say that a proprtion around ɸ might be preferred by many but the popular range for aspect ratio b/a of objects is more like ɸ ± 20%.  Similarly, claims about proprtion of human body parts etc. do not stand up to scrutiny.  
However, an aspect ratio based around the golden ratio may have some truth. I shall stick my neck out here and say that when we look at the view in front of us, we see a larger horizontal span but the size of our vertical view is truncated by the ground we are standing on.  This trains our mind to function best when we have a view that is about 1.5 times wider than it is higher - and it is possible that a preference of this kind of aspect ratio might be hard-wired in our aesthetics. 

Fibonacci Spiral:  Also referred to as a Golden Spiral and a source of untold confusion in popular science articles.

If you google 'spirals in nature and design', you will find a large collection of articles about discovering spiral patterns in nature and art with some very nice pictures too.  Mostly it is the logarithmic spiral that one observes but in many publications it is claimed that what we are seeing is the golden spiral with every 90 degrees turn, the size of the spiral increases by the golden ratio (1.618033...).  This generally does not stand up to scrutiny - for instance, a popular example is the size of the nautilus shell.  The shell does grow as a logarithmic spiral but the growth per ninety degree turn is not the golden ratio (1.618033...) but has been measured to range from 1.33 ± 20%  to 1.7.  

I find the logarithmic spiral fascinating and shall discuss it in more detail here.  First thing to note is that Fibonacci spiral has constant 'local' curvature (quarter circles in successive squares) while a logarithmic spiral (golden spiral is a special case of it) has a continuously varying curvature. 
The next slides list some interesting properties and examples of logarithmic spirals:

Did you know that a peregrine falcon, while hunting,  swoops at speeds up to 220 miles per hour (~360 km/hour) - fastest speed of any animal in the world!! 

Golden Angle:  If we divide the circumference of a circle in two parts according to Euclid's prescription (effectively, change the straight line to make the perimeter of a circle) then the smaller of the two angles is called the Golden Angle.

The question is:  For a new primodium to start,  the plant must know where the least crowded spot on the meristem is?  The best location seems to be situated at an angle equal to the golden angle from the last primordium - but how does the plant know this? This is where it gets technical - A full description of phyllotaxis is not yet established and lot of questions remain.   I shall give a  very brief summary. 
Molecular-genetic experiments indicate that active transport of the plant hormone auxin is the key process regulating phyllotaxis.  Auxin is a plant hormone produced in the stem tip that promotes cell elongation.  Organ primordia produce an inhibitory field - depleted local auxin levels - that prevents organ initiation within a certain proximity.
Mechanical stresses from cell deformation at the site of primordia are also hypothesized to play an important role.  
Locating new primordia at exact separation of 137.5 degrees is not understood although some attempts have been made on the basis of packing seeds in a limited space.  

Regular Pentagon and The Petagram:  I discuss these as a pentagram has some unique mathematical properties and the sides and angles of a pentagram are intimately connected to the golden ratio.  Because of its symmetry, a pentagram has been attractive to mathmaticians, designers etc.

Final Note:  I have really enjoyed working on this blog article.  The Golden Ratio is an amazing number and I only wish that people do not hype its relevance to nature and humans too much.  It is a pure geometrical delight and its close relation to Fibonacci Numbers is a great surprise to me - that is why mathematics is so much fun - great feeling of wonder and very satisfying when you have completed a proof.
I am not finished with Phyllotaxis and hope to return to it sometime in future - first I need to find a friendly botanist in the University with some spare time!
By the way, you can look here to see how they even find Golden Ratio and Fibonacci Numbers in the stock market - almost as fantastic as our theology friends.

PS:  The golden ratio springs up in unexpected places - a popular maths puzzle is   6ܑⁿ + 4ⁿ = 9ⁿ.  The value of n is actually related to the inverse of the golden ratio. Amazing!  

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