File:Stokes drift 1d.gif

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Summary

DescriptionStokes drift 1d.gif	English: Stokes drift in a one-dimensional compressible flow, with the red line the exact solution and the blue line the 2nd-order perturbation-series solution. The Stokes drift $\xi (\xi _{0},t)$ is the solution of the partial differential equation: ${\frac {\partial \xi }{\partial t}}={\hat {u}}\,\cos(k\xi -\omega t),$ with ${\hat {u}}\,\cos(k\xi -\omega t)$ the Eulerian flow velocity as a function of position $\xi$ and time $t.$ Further $k$ is the wavenumber and $\omega$ is the angular frequency. The animation shows the evolution in time $t$ of $\xi (\xi _{0},t)-\xi _{0}$ as a function of position label $\xi _{0}.$ In the presented case the values of the parameters are: $\omega =k=1$ and ${\hat {u}}=0.3.$ The red dots show Lagrangian drifter positions $x=\xi (\xi _{0},t)$ for equidistant labels $\xi _{0}$ evolving with time. The example is analogue to the one described in: G. Falkovich (2011). Fluid Mechanics (A short course for physicists). Cambridge University Press. ISBN 978-1-107-00575-4, pp. 71–72.
Date	7 March 2017
Source	Own work
Author	Kraaiennest
Background InfoField	The exact solution fo a progressive wave of permanent form is: $\xi ={\frac {2}{k}}\arctan \,\left({\frac {\sigma }{\omega +k{\hat {u}}}}\,\tan({\tfrac {1}{2}}kx-{\tfrac {1}{2}}\sigma t)\right)+{\frac {\omega t+n\,2\pi }{k}},$ where $\sigma =\omega \,{\sqrt {1-\left({\frac {k{\hat {u}}}{\omega }}\right)^{2}}}\qquad {\text{and}}\qquad n=\operatorname {round} \left({\frac {kx-\sigma t}{2\pi }}\right),$ with the round function denoting rounding to the nearest integer. The corresponding second-order perturbation series solution shown in the animation is: $\xi =\xi _{0}-{\frac {\hat {u}}{\omega }}\,\sin(k\xi _{0}-\omega t)+{\tfrac {1}{4}}\,{\frac {k{\hat {u}}^{2}}{\omega ^{2}}}\,\sin(2k\xi _{0}-2\omega t)+{\tfrac {1}{2}}\,{\frac {k{\hat {u}}^{2}}{\omega ^{2}}}\,t+{\mathcal {O}}\left(\left({\frac {k{\hat {u}}}{\omega }}\right)^{3}\right).$

This diagram was created with MATLAB.

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File history

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	Date/Time	Dimensions	User	Comment
current	22:06, 1 May 2017	640 × 240 (1.04 MB)	Kraaiennest	correct vertical axis label
	21:53, 1 May 2017	640 × 240 (1.03 MB)	Kraaiennest	show also Lagrangian drifter positions; reduce file size
	20:02, 7 March 2017	620 × 233 (1.7 MB)	Kraaiennest	improve colors
	20:01, 7 March 2017	620 × 233 (1.22 MB)	Kraaiennest	filesize reduction
	19:40, 7 March 2017	620 × 233 (4.53 MB)	Kraaiennest	smaller size
	19:27, 7 March 2017	1,280 × 480 (3.86 MB)	Kraaiennest	User created page with UploadWizard