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10: Postface

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    We have developed skills in mathematics and advanced scientific reasoning that allow us to take a set of assumptions (hypotheses) and develop from them a testable prediction in the form of a set of partial differential equations. The result of this is the Navier-Stokes equations and the accompanying mass equation, heat equation, etc.

    To test these predictions is not easy, for one must not only solve the equations but measure a real flow with precision sufficient to tell whether the solution matches reality or not. We have played at this by extracting some extremely simple solutions for idealized model flows and comparing them with reality on scales that we can perceive easily without specialized equipment. In some cases, the match is good. When it is not, the culprit has most commonly been a failure to account for turbulence. This is not a failing of the Navier-Stokes equations but rather of the unrealistically simple flow geometries for which we are able to solve them.

    Even in nature, where flows are invariably turbulent, we see things like waves, vortices, and hydraulic jumps, and they behave much as our simple idealizations predict. If predicted scales are seriously inaccurate, inserting a simple model of the turbulent energy cascade often gives realistic results.

    So where do we go from here? The student of oceanography will go on to study the effects of density stratification and planetary rotation. The atmospheric physicist will need these and also the thermodynamics of water vapor. Plasmas found in the ionosphere and stellar atmospheres may be understood by adding Maxwell’s equations for electromagnetism (Choudhuri 1996). In smaller systems like lakes, rivers and beaches, often of concern to civil and environmental engineers, density stratification is important but planetary rotation is less so. In all geophysical systems, turbulence must be accounted for to achieve a realistic level of understanding and a predictive capacity.

    For further exploration, check out Fluid Mechanics (Kundu et al. 2016). It contains concise summaries of most of these advanced aspects of the discipline and many more (and I do mean many) not listed here.

    As you walk through the world, you are surrounded by flow. You contain flow. You now carry with you a conceptual understanding developed over centuries by humans like you who also walked through, and were part of, this world of flow. Remember the words of Bruce Lee: Be water, my friend.

    Bill Smyth


    Aris, R., 1962: Vectors, Tensors and the Basic Equations of Fluid Mechanics. Cambridge Univ. Press.

    Bronson, R. and G. Costa, 2009: Matrix Methods: Applied Linear Algebra (3rd ed.). Academic Press, New York.

    Choudhuri, A., 1996: The Physics of Fluids and Plasmas, an Introduction for Astrophysicists. Cambridge Univ. Press.

    Curry, J. A. and P. J. Webster, 1998: Thermodynamics of atmospheres and oceans, volume 65. Academic Press.

    Gill, A., 1982: Atmosphere-Ocean Dynamics. Academic Press, San Diego„ 662 pp.

    Kundu, P., I. Cohen, and D. Dowling, 2016: Fluid Mechanics (6th ed.). Academic Press, San Diego.

    Li, C., A. Valle-Levinson, L. Atkinson, K. Wong, and K. Lwiza, 2004: Estimation of drag coefficient in James River Estuary using tidal velocity data from a vessel-towed ADCP. J. Geophys. Res.. J. Geophys. Res., 109, doi:10.1029/2003JC001991, c03034.

    Marion, J., 2013: Classical dynamics of particles and systems. Academic Press, New York.

    Moffatt, H., S. Kida, and K. Ohkitani, 1994: Stretched vortices - the sinews of turbulence; large reynolds number asymptotics. J. Fluid Mech., 259, 241–264.

    Smyth, W., 1999: Dissipation range geometry and scalar spectra in sheared, stratified turbulence. J. Fluid Mech., 401, 209–242.

    Smyth, W. and S. Thorpe, 2012: Glider measurements of overturning in a kelvin-helmholtz billow train. J. Mar. Res., 70, 119–140.

    Vallis, G., 2006: Atmospheric and Oceanic Fluid Dynamics. Cambridge Univ. Press.

    White, F., 2003: Fluid Mechanics. McGraw-Hill, New York.

    10: Postface is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Bill Smyth via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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