# Re: Calculating stream function from known velocity field.

I think there is some confusion about streamlines, streamlines and streaklines.  I also think that what has been written about them in this forum is incomplete, and this can lead to some misconceptions and erroneous results.

G. K. Batchelor's book, An Introduction to Fluid Dynamics, 1967, Cambridge University Press, is a good reference.

A streamline is a line everywhere tangent to the velocity field at an instant in time.  It's equation is dx/u = dy/v = dz/w.

A path line is the path of a material element of fluid, i.e. the Lagrangian drifter path.  One would integrate along a fluid particle path in time to compute its path line.

A streak line is the line on which lie all those fluid elements that at some earlier instant passed through a certain point in space, e.g. a point where dye is injected.

For steady flow, streamlines, path lines and streak lines coincide, but in general they do not.  Unless Ansley's routines integrate in time, they cannot be computing streak lines (unless steady flow is assumed, and then they calculate all three).

Billy Kessler gave formulae for computing the two-dimensional stream function which applies when the flow is horizontally nondivergent, i.e. when dw/dz = 0 everywhere.  This is rarely true in geophysical flows.  For general oceanographic and atmospheric problems, an assumption like this and diagrams of the supposed two-dimensional stream function that satisfied those equations would be misleading.  Even though the vertical velocity, w, is small its gradient needs to be considered.

If Ansley's routines compute the streamfunction, is it the two-dimensional streamfunction that assumes dw/dz = 0?

--
Edward D. (Ned) Cokelet, Ph.D.      cokelet@pmel.noaa.gov
NOAA/PMEL                                   ph: (206) 526-6820
7600 Sand Point Way NE                   fax: (206) 526-6485
Seattle, WA 98115-6439

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