Category Archives: nonlinear dynamics

I found a great dune paper, but too late

There it was. The last figure in Ritchie and Penland (1988). A conceptual model for coastal dune development on barrier islands subject to storms. It is the figure I was looking for in 2015 when I wrote a paper on coastal dunes — I just found it one year too late..

Here is the figure from Ritchie and Penland (1988):


The figure lays out a tidy conceptual model where coastal dunes tend to develop with time, subject to erosion from minor storm events. After some period of time, the minor storms stop causing dune erosion and the dune reaches a final stage of development before being totally destroyed by a major storm.

Now to my own figure in Goldstein and Moore (2016):GM.jpg

Which shows the trajectory of the nondimensional dune height (D*) as the dune is subjected to periodic storm events (with different characteristics, hence the multiple curves). This is not the exact as from Ritchie and Penland, but it sure is close (and I could certainly have made a more similar version).

This sort of issue — discovering a great reference too late — is bound to happen for coastal dune researchers (since so many coastal dune papers are published) and for other earth scientists since so many papers are published. Preliminary work suggests that there may not be a major growth in reference section size. So I bet others have this problem of finding a good paper too late  — but it still hurts.

three types of analogy

What does it mean for a system to have an ‘analog’ or have an ‘analogous’ counterpart? A simple description could be: an analog of system ‘A’ shares some similarity in dynamics to another system ‘B’. We can profit from an analogous system (an ‘analog’) by knowing something about ‘A’ and seeing if this leads to new insight about ‘B’ — ideally allowing us to make predictions about system ‘B’ from system ‘A’.

A first type of analogy is when ‘A’ and ‘B’ are the same system, but at different times.  This is a data-driven way of investigating a system based on the previous observed states of the same system (i.e., ‘it looked like this last week, and it rained, so I bet it will rain’). An example of this way of thinking was presented by Lorenz (1969a, 1969b).

A second type of analogy is two different descriptions of the same system. For example, a numerical model of a sand dune is just a low-dimensional analogy of a real world sand dune. The behavior of the model might not reproduce the dynamics of the real world sand dune (think of all the individual sand grains! all those degrees of freedom!), but the basic properties of a sand dune are reproduced.

A third analogy is when two different systems resemble one another. A perfect example of this is the Lorenz equations and the chaotic water wheel, where the mathematical description of the systems are identical. I assume other such examples exist even when the systems are not perfect mathematical equivalents. Less strict or confined analogs can be useful —  a system may ‘looks like’ or have similarities to another.

I have been thinking about the nature of analogy after reading Stephon Alexander’s book ‘The Jazz of Physics’. Dr. Alexander uses jazz (and ideas from jazz) as analogies to explore ideas in cosmology — and it seems like working with analogies has yielded some success for him.



  • Lorenz, E.N., 1969a, Atmospheric predictability as revealed by naturally occurring analogues. J Atmos Sci 26: 636–646.
  • Lorenz, E.N., 1969b, Three approaches to atmospheric predictability. Bull Am Meteorol 50: 345–349.

Also see this recent paper, which is a neat example of analogy: