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Talk:Stokes drift

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1D example of Falkovich

[edit]
Animation over time of the Lagrangian parcel position as a function of The red line is the exact solution and the blue line is the 2nd-order perturbation-series solution. The parameters are and Note the slightly larger Stokes drift velocity and lower phase speed of the exact solution.
The red dots show Lagrangian drifter positions for equidistant labels evolving with time.

Note that the Stokes drift in Falkovich' example of 1D flow has an exact solution. In this case, the Eulerian velocity is taken as – where instead of the sine as used by Falkovich, the cosine is used because of symmetry conditions of at and Now the Lagrangian parcel position is denoted as with the position label taken equal to The position is the solution of:

The additional condition on is that at the Stokes drift is equal to zero, i.e. that the spatial mean value of the oscillation is zero: Then the progressive wave solution is:

where

with the round function denoting rounding to the nearest integer.

It can directly be observed that the Lagrangian moving parcel experiences a different (lower) frequency than the Eulerian velocity frequency The Stokes drift velocity is simply the difference in positions after one Lagrangian wave period has passed, divided by the Lagrangian wave period. So the exact expression for the Stokes drift velocity is:

It has the Taylor expansion:

in agreement with Falkovich' perturbation solution. Which is in this case – with a cosine for the velocity field, :

-- Crowsnest (talk) 15:29, 6 March 2017 (UTC)[reply]