What do Jupiter’s Great Red Spot, the Gulf Oil Spill and nutrient patterns in the ocean have in common? It is the seemingly incompatible diversity inherent in this question that first drew me to the area of applied mathematics known as Lagrangian Coherent Structures (LCS). LCS are most intuitively described in terms of familiar fluid structures from nature: hurricane centres, ocean gyres and atmospheric eddies. They are Lagrangian because they are dependent on initial conditions; their particular structure may differ depending on the time-slice you look at. They are coherent because they possess ‘exceptional’ properties in comparison to surrounding fluid. It is defining what we mean by ‘exceptional’ that presents the interesting mathematical challenge I pursued during my research project. Looking at a satellite image of a hurricane, most of us could point out the hurricane. But how could we mathematically describe that hurricane’s structure?

There are two intriguing aspects to defining coherence in fluid structures. One of these arises from the well-established mathematics of steady, or time-independent, fluid flows. There are quite simple techniques available to identify the structures present in steady flows. However, add time-dependence and things become significantly more complex. The structures we seek are now dynamic entities which may evolve over time, so our definition of coherence must allow for ‘coherent time-evolution’ while also agreeing with the established mathematics of steady flow techniques. The second issue arises from the diversity of structures we must describe. We desire a sufficiently general definition to encompass the important fluid structures we observe in nature.

One way to approach this intimidating generality is to consider two key properties of fluid structures, rotational and strain. Rotational structures are regions of exceptional fluid rotation, such as in hurricane cores. Strain structures are regions of high stretching, such as in rip tide currents. In my project, guided by my supervisor Assoc. Prof. Sanjeeva Balasuriya, I investigated two techniques, the Finite-Time Lyaponov Exponent, capable of extracting strain structures, and the Lagrangian-Averaged Vorticity Deviation, capable of extracting rotational structures. Most flows possess both rotational and strain structures. Comparing structures identified by these two techniques thus allows significant insight into a flow’s structure.

Completing this project, one of the learning experiences I most appreciated was how mathematics provides a framework for understanding the subtle similarities between diverse physical systems. Before starting my research I probably would have answered ‘Nothing!’ to my question in the first line of this blog. My brief foray into this fascinating area of applied mathematics has only skimmed the surface of a vast body of research, but I feel I now have a better appreciation of incredible natural structures like Jupiter’s Great Red Spot than previously.  I am definitely curious to further pursue the exploration of Earth’s dynamical systems using mathematics.

1. NASA/JPL/Space Science Institute, available at https://www.nasa.gov
2. Rev. Fluid Mech., 2015, 47:137–62.
3. SeaWiFS Project, NASA/Goddard Space Flight Center, and ORBIMAGE, available at https://earthobservatory.nasa.gov

Rose Crocker was one of the recipients of a 2017/18 AMSI Vacation Research Scholarship.