Thus, for example, the Thurston–Nielsen classification has been fruitfully applied to the problem of stirring in two-dimensions by any number of stirrers following a time-periodic 'stirring protocol' (Boyland, Aref & Stremler 2000).
Such structures are characterised at least in part by the helicity of certain sub-regions of the flow field, a topological invariant of the equations.
It is known that, with few exceptions, any statistically homogeneous turbulent flow having nonzero mean helicity in a sufficiently large expanse of conducting fluid will generate a large-scale magnetic field through dynamo action.
A mathematical introduction to this subject is given by Arnold & Khesin (1998) and recent survey articles and contributions may be found in Ricca (2009), and Moffatt, Bajer & Kimura (2013).
Topology is also crucial to the structure of neutral surfaces in a fluid (such as the ocean) where the equation of state nonlinearly depends on multiple components (e.g. salinity and heat).