Topological ideas are relevant to fluid dynamics (including

magnetohydrodynamics Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...

) at the

kinematic Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fiel ...

level, since any fluid flow involves continuous deformation of any transported scalar or vector field. Problems of stirring and mixing are particularly susceptible to topological techniques. 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). Other studies are concerned with flows having chaotic particle paths, and associated exponential rates of mixing (Ottino 1989). At the dynamic level, the fact that vortex lines are transported by any flow governed by the classical

Euler equations 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...

implies conservation of any vortical structure within the flow. 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. Helicity plays a central role in

dynamo theory In physics, the dynamo theory proposes a mechanism by which a celestial body such as Earth or a star generates a magnetic field. The dynamo theory describes the process through which a rotating, convecting, and electrically conducting fluid can ...

, the theory of spontaneous generation of magnetic fields in stars and planets (Moffatt 1978, Parker 1979, Krause & Rädler 1980). 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. Such fields themselves exhibit

magnetic helicity In plasma physics, magnetic helicity is a measure of the linkage, twist, and writhe of a magnetic field. In ideal magnetohydrodynamics, magnetic helicity is conserved. When a magnetic field contains magnetic helicity, it tends to form large-scal ...

, reflecting their own topologically nontrivial structure. Much interest attaches to the determination of states of minimum energy, subject to prescribed topology. Many problems of fluid dynamics and

fall within this category. Recent developments in topological fluid dynamics include also applications to magnetic

braids A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...

in the

solar corona A corona ( coronas or coronae) is the outermost layer of a star's atmosphere. It consists of plasma. The Sun's corona lies above the chromosphere and extends millions of kilometres into outer space. It is most easily seen during a total solar ...

, DNA knotting by

topoisomerases DNA topoisomerases (or topoisomerases) are enzymes that catalyze changes in the topological state of DNA, interconverting relaxed and supercoiled forms, linked (catenated) and unlinked species, and knotted and unknotted DNA. Topological issues i ...

, polymer entanglement in chemical physics and chaotic behavior in dynamical systems. 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). Fluid parcels remain neutrally

buoyant Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pr ...

as they move along neutral surfaces, despite variations in salinity or heat. On such surfaces, the salinity and heat are functionally related, but this function is multivalued. The spatial regions within which this function becomes single-valued are those where there is at most one

contour Contour may refer to: * Contour (linguistics), a phonetic sound * Pitch contour * Contour (camera system), a 3D digital camera system * Contour, the KDE Plasma 4 interface for tablet devices * Contour line, a curve along which the function ha ...

of salinity (or heat) per isovalue, which are precisely the regions associated with each edge of the Reeb graph of the salinity (or heat) on the surface (Stanley 2019).

References

* Arnold, V. I. & Khesin, B. A. (1998) ''[ftp://84.237.21.152/pub_archive/kolhoz/M_Mathematics/MP_Mathematical%20physics/Arnold%20V.,%20Khesin%20B.%20Topological%20Methods%20in%20Hydrodynamics%20(ISBN%20038794947x)(Springer,%201998)(393s).pdf Topological Methods in Hydrodynamics]''. Applied Mathematical Sciences 125, Springer-Verlag. *Boyland, P.L., Hassan Aref, Aref, H. & Stremler, M.A. (2000
Topological fluid mechanics of stirring
''J.Fluid Mech.'' 403, pp. 277–304. *Krause, F. & Rädler, K.-H. (1980) ''Mean-field Magnetohydrodynamic and Dynamo Theory''. Pergamon Press, Oxford. * Moffatt, H.K. (1978
''Magnetic Field Generation in Electrically Conducting Fluids''
Cambridge Univ. Press. * Moffatt, H.K., Bajer, K., & Kimura, Y. (Eds.) (2013
''Topological Fluid Dynamics, Theory and Applications''
Kluwer. *Ottino, J. (1989
''The Kinematics of Mixing: Stretching, Chaos and Transport''
Cambridge Univ. Press. * Parker, E.N. (1979) ''Cosmical Magnetic Fields: their Origin and their Activity''. Oxford Univ. Press. * Ricca, R.L. (Ed.) (2009
''Lectures on Topological Fluid Mechanics''
Springer-CIME Lecture Notes in Mathematics 1973. Springer-Verlag. Heidelberg, Germany. {{ISBN, 9783642008368 *Stanley, G. J., 2019
Neutral surface topology
Ocean Modelling 138, 88–106. Fluid dynamics Topological dynamics