[Coral-List] Geometry, coral bands, and tree rings

[John Ware]

  Dear List,

I suppose that what I am going to provide has been done to death in many 
places, but here goes some simple geometric considerations.  There will 
be a bunch of simplifying assumptions, but I doubt that making the 
problem more complex will change the conclusions much.

Suppose we have a hemispherical coral colony with each polyp able to 
deposit a fixed amount of CaCO3 each year.  This increases the volume of 
the coral head and thus its area.  Increasing the area allows for more 
polyps to occupy the space.  And so on.  The process is, of course, 
non-linear.  So I wrote a little computer program that simulates the 
process.  

I picked some numbers that I thought were reasonable and that resulted 
in a 10 cm coral growing to a diameter of 3.2 m in 500 yrs and 6.2 m in 
1000 yrs.   What is interesting about this little experiment is that the 
coral diameter increase is NOT linear with time.  The rate of change of 
diameter (and hence the spacing between bands) actually increases each 
year.  HOWEVER, the rate of change is very small.  The increase in 
diameter at 100 yrs is 0.610 cm/yr and at 1000 yrs is 0.615 cm/yr.

I find similar results across a range of assumed growth rates.

The rate of change is probably much too small to be detected leading to 
the conclusion that bands are about of equal width throughout the life 
of the coral.

Now trees are a little different.  There are lots of things that can act 
to evolve various growth strategies (competition, bending stress in 
winds, and so on).  However, it is not unreasonable to assume that the 
amount of energy that can be devoted to growth is proportional to crown 
area and that trunk area is proportional to crown area.  If some 
fraction of the energy absorbed each year from radiation goes into 
increasing the cross-sectional area of the trunk, then this area will 
increase linearly.  This will result in the radius increasing as the 
square root of the area with the result that growth rings will get 
closer and closer together.

Well, I don't suppose I did anything particularly new or clever, but I 
thought I would throw it out there anyway.

John

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    *                      John R. Ware, PhD                    *
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