# Wind vector transformation from Gaussian grid to displaced pole grid

I have been given the "u" and "v" component with respect to an earth coordinate reference system(Gaussian grid - https://en.wikipedia.org/wiki/Gaussian_grid http://www.nco.ncep.noaa.gov/pmb/docs/on388/tableb.html#GRID98) in terms of latitude and longitude and I need to transform this pair of vector into a new coordinate system whose north pole is defined not relative to the earth's north pole but relative to a "imaginary" north pole. The Gaussian grid is not defined at the poles due to singularities.

What I do believe is that the wind vector pair must transform together(as shown below) but I am not sure about the transformation and how to go about calculating it.

uNew = u*cos(rot) - v*sin(rot)

vNew = u*sin(rot) + v*cos(rot)


A combined wind vector needs to be obtained by first calculating the magnitude and the direction. Then the components of this wind vector need to be rotated into the new coordinate frame.

Following Verkley(1984) for any known position of the north pole ($\lambda_0$,$\theta_0$) in the unrotated coordinate system the following identities specify the transformation between the unrotated coordinates and rotated coordinates. The following two appear from the law of cosines

$\sin\phi'$ = $\sin\phi$ $\sin\phi_0$ + $\cos\phi$ $\cos\phi_0$ $\cos(\lambda-\lambda_0$)

This is the inverse transformation

$\sin\phi$ = $\sin\phi'$ $\sin\phi_0$ - $\cos\phi'$ $\cos\phi_0$ $\cos\lambda'$

From the law of sines

$\cos\phi'$ $\sin\lambda'$ = $\cos\phi$ $\sin(\lambda-\lambda_0)$

As input I have been given velocities(zonal and meridional) in the geographical coordinate system, latitude and longitude of rotated lat lon grid(latitude and longitude of the rotated north pole) and the latitude and longitude of a point in the geographical coordinate system. How do I obtain the velocities in the rotated frame ?

• Could you provide the formula used to transform points? – Emanuele Paolini Dec 20 '15 at 5:57
• @EmanuelePaolini yes I believe so. I have a reference that shows that. I will make the edit and let you know. – gansub Dec 20 '15 at 6:04
• @EmanuelePaolini - I added the formula used to transform points – gansub Dec 20 '15 at 14:20