Re: [ferret_users] Vertical velocity in Soda3.15.2 monthly mean data

[Satheesh Shenoi <shenoi1958@xxxxxxxxx>]

Dear Ryo,

A million thanks for the detailed explanation. It was very helpful to me and maybe to the other users of Ferret as well. 

It is clear that it is nearly impossible to get the correct w velocities using the data remapped on a non-native model grid. In my case, the  SODA data was already remapped from the varying tripolar Arakawa-B grids to a uniform 1/2x1/2 Mercator coordinate horizontal grid. So, I may have to check with the SODA team whether they can generate the w field using the original model outputs. I will keep you posted.

with best regards

Satheesh

Email: shenoi1958@xxxxxxxxx

Mobile: +91 9441013377

On Fri, 1 Sept 2023 at 20:28, Ryo Furue <furue@xxxxxxxxxx> wrote:

Dear Satheesh and Ferret users,

On Fri, Sep 1, 2023 at 6:16 PM Satheesh Shenoi <shenoi1958@xxxxxxxxx> wrote:

[ . . . ]

If I understand your response correctly, it is possible to get the vertical velocity using the continuity equation by smoothing 'w' appreciably. 

That's not quite what I meant.  I meant:

1. Use exactly the same discrete continuity equation that the model uses.  Different discretization will often result in large error.

2. If you cannot use the exact discrete continuity equation, then you would need to heavily smooth the w field calculated with an ad-hoc discrete continuity equation.

In case 2, you don't know how much smoothing is appropriate because you don't know the true w field. 

I would like to try it out because my earlier attempt without non-smoothing did not yield any reasonable structure for w.

 

However, I have a question because you say that because your numerical code uses B-grid and the Cartesian coordinates, u[x=@DDC] + v[y=@DDC] is the right calculation. 

On the B-grid, both velocity components, u and v, are located at the same gridpoint.  So, both central-difference derivatives u[x=@DDC] and v[y=@DDC] are located at the same velocity points.  In the C-grid, the u and v components are located at different grid points and there u[x=@DDC] and v[y=@DDC] are located at different gridpoints.  For this reason, to sum the discrete ∂u/∂x and ∂v/∂y, you need to map them to a common grid by interpolation.

With SODA, you use the spherical coordinate.  Then,

div(u, v) = (∂u/∂λ)/(r cos ϕ) + [∂(v cos ϕ)/∂ϕ] / (r cos ϕ)  . . . (A)

(See https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates ).  @DDC calculates the correct "gradient" in both directions in the spherical coordinates.  So, the zonal gradient (∂u/∂λ)/(r cos ϕ) is correct because dx = r (cos ϕ) dλ.  On the other hand, the meridional "gradient" is

 (∂v/∂ϕ) / r . . . (B)

because dy = r dϕ .  This gradient is, however, not the derivative we want: compare (B) with (A).  Normally, this error is minor unless you go to high latitudes, but, as I demonstrated in my previous message, a small error in horizontal divergence translates to large error in w . . .  I don't know whether this error would be cured by smoothing . . .

The GCM of course uses a discrete version of (A), which isn't hard to program if you know how to do it . . . 

Mercator coordinate horizontal grid

Oops!  I'm not familiar with that coordinate system.

Perhaps the best way may be to ask the creators of SODA.

Ryo