Category Archives: nonlinear dynamics

Nonlinear Dynamics and Geomorphology

This is a list of geomorphology papers that map onto the chapter headings of Strogatz — ‘Nonlinear Dynamics and Chaos’. This is a work in progress — some headings are left blank because I can’t find concrete examples (i.e., strange attractors) and while others remain blank because of too many examples (i.e., fractals). I envision this list could be used when teaching or discussing nonlinear dynamics in a geomorphology setting.

  • Part I: One-Dimensional Flows
    • Ch. 2: Flows on a Line
    • Ch. 3: Bifurcations
      • Fagherazzi, S., Carniello, L., D’Alpaos, L., & Defina, A. (2006). Critical bifurcation of shallow microtidal landforms in tidal flats and salt marshes. Proceedings of the National Academy of Sciences, 103(22), 8337-8341. 10.1073/pnas.0508379103
      • Anderson, R. S. (2002). Modeling the tor-dotted crests, bedrock edges, and parabolic profiles of high alpine surfaces of the Wind River Range, Wyoming. Geomorphology, 46(1), 35-58. 10.1016/S0169-555X(02)00053-3
      • Pelak, N. F., Parolari, A. J., & Porporato, A. (2016). Bistable plant–soil dynamics and biogenic controls on the soil production function. Earth Surface Processes and Landforms, 41(8), 1011-1017.10.1002/esp.3878
      • Yizhaq, H., Ashkenazy, Y., & Tsoar, H. (2007). Why do active and stabilized dunes coexist under the same climatic conditions?. Physical Review Letters, 98(18), 188001. 10.1103/PhysRevLett.98.188001
      • Yizhaq, H., Ashkenazy, Y., & Tsoar, H. (2009). Sand dune dynamics and climate change: A modeling approach. Journal of Geophysical Research: Earth Surface, 114(F1). 10.1029/2008JF001138
      • Bel, G., & Ashkenazy, Y. (2014). The effects of psammophilous plants on sand dune dynamics. Journal of Geophysical Research: Earth Surface, 119(7), 1636-1650. 10.1002/2014JF003170
      • Goldstein, E.B., and L.J. Moore, (2016) Stability and bistability in a one-dimensional model of coastal foredune height, J. Geophys. Res. Earth Surf.121964977doi: 10.1002/2015JF003783
    • Ch. 4: Flows on a Circle
  • Part II: Two-Dimensional Flows
    • Ch. 5: Linear Systems
      • Plant, N. G., Todd Holland, K., & Holman, R. A. (2006). A dynamical attractor governs beach response to storms. Geophysical Research Letters, 33(17). 10.1029/2006GL027105
    • Ch. 6: Phase Plane
      • Marani, M., D’Alpaos, A., Lanzoni, S., Carniello, L., & Rinaldo, A. (2007). Biologically‐controlled multiple equilibria of tidal landforms and the fate of the Venice lagoon. Geophysical Research Letters, 34(11).10.1029/2007GL030178
      • Marani, M., D’Alpaos, A., Lanzoni, S., Carniello, L., & Rinaldo, A. (2010). The importance of being coupled: Stable states and catastrophic shifts in tidal biomorphodynamics. Journal of Geophysical Research: Earth Surface, 115(F4). 10.1029/2009JF001600
      • Stark, C. P., & Passalacqua, P. (2014). A dynamical system model of eco‐geomorphic response to landslide disturbance. Water Resources Research, 50(10), 8216-8226.10.1002/2013WR014810
      • Stark, C. P. (2006), A self-regulating model of bedrock river channel geometry, Geophys. Res. Lett., 32, L04402, doi:10.1029/2005GL023193.
      • Limber, P. W., A.B. Murray, P. N. Adams and E.B. Goldstein, (2014), Unraveling the dynamics that scale cross-shore headland amplitude on rocky coastlines, Part 1: Model Development,Journal of Geophysical Research: Earth Surface, 119, doi: 10.1002/2013JF002950
      • Limber, P. W., & Murray, A. B. (2014). Unraveling the dynamics that scale cross‐shore headland relief on rocky coastlines: 2. Model predictions and initial tests. Journal of Geophysical Research: Earth Surface, 119(4), 874-891.10.1002/2013JF002978
      • Mariotti, G., & Fagherazzi, S. (2013). Critical width of tidal flats triggers marsh collapse in the absence of sea-level rise. Proceedings of the National Academy of Sciences, 110(14), 5353-5356. 10.1073/pnas.1219600110
    • Ch. 7: Limit Cycles
      • Stark, C. P. (2010). Oscillatory motion of drainage divides. Geophysical Research Letters, 37(4).10.1029/2009GL040851
    • Ch. 8: Bifurcations revisited
      • Mariotti, G., & Fagherazzi, S. (2013). A two‐point dynamic model for the coupled evolution of channels and tidal flats. Journal of Geophysical Research: Earth Surface, 118(3), 1387-1399. 10.1002/jgrf.20070
  • Part III: Chaos
    • Ch. 9: Lorenz Equations
    • Ch. 10: One-Dimensional Maps
      • Goldstein, E.B., and L.J. Moore, (2016) Stability and bistability in a one-dimensional model of coastal foredune height, J. Geophys. Res. Earth Surf.121964977doi: 10.1002/2015JF003783
    • Ch. 11: Fractals
      • There are too many papers/books/issues to discuss here…
    • Ch. 12: Strange Attractors

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: