Re: [ferret_users] how to calculate the streamfunction and velocity potential

["William S. Kessler" <William.S.Kessler@xxxxxxxx>]

Yes, thank you Ryo. I should have said that. The streamfunction  
exists only when the flow is nondivergent, typically when u and v are  
integrated from the ocean bottom to the surface.

Another use is a vertical streamfunction, valid for a zonal, coast-to- 
coast integral. That is easier to calculate, because w is identically  
zero at the surface:

let streamfn=v[x=@din,z=@iin]   ! where v is in a single, bounded basin

And a velocity potential is valid only when the flow is irrotational  
(curl(v)=0), which is unlikely to be the case in the ocean. In the  
atmosphere, it can be used to describe the upper-level circulation  
which is predominantly zonal.

Billy K

On Aug 8, 2007, at 6:46 PM, Ryo Furue wrote:

> Hi Billy and Hanh,
>
> | ! compute the horizontal streamfunction from vertically integrated u
> | ! and v fields
> | ! note that u and v must occur on the same spatial points (B-grid)
> | ! arg1=variable name for u
> | ! arg2=variable name for v
> | ! arg1=x0 (must be on a gridpoint or set mode interp)
> | ! arg2=y0 (likewise)
> | ! psi=int(y0,y)u(x,y)dy - int(x0,x)v(x,y0)dx
> | ! u=d(psi)/dy, v=-d(psi)/dx
>
> I guess this gives you a correct streamfunction only
> when du/dx + dv/dy = 0, right?  (There's no function satisfying
> both u=d(psi)/dy and v=-d(psi)/dx when du/dx + dv/dy is nonzero.)
>
> Hanh mentioned velocity potential, which suggests that a Helmholz
> decomposition is required:
>
>   (u,v) = grad(phi) + k x grad(psi)
>
> To obtain phi(x,y) and psi(x,y), you need to solve these Poisson  
> equations
>
>   div grad (phi) = div(u,v)
>   div grad (psi) = curl(u,v)
>
> with some (probably arbitrary) boundary conditions.
>
> I myself haven't done this, but a colleague of mine uses the method
> described by:
>
>   Watterson, I., 2001. Decomposition of global ocean currents using a
>   simple iterative method. J. Atmos. Oceanic Tech., 18, 691–703.
>
> Hope this helps,
> Ryo