Equations in the rotated grid

The equations need some modifications for use in our rotated reference frame, The changes are detailed below.

Curvature term

The curvature term compensates for the equations being derived on the flat but the World being round. The term is the same form as before,

(A + u) cos(θ') sin(φ')

but the direction in the rotated frame, θ' and the latitude in the rotated frame φ' are used in place of θ and φ.

Vorticity

The real length covered by a change in longitude is R_E cos φ dλ, where R_E is the radius of the Earth, φ is latitude and λ is longitude. And the real length covered by a change in latitude is R_E dφ. Hence the real voriticity is given by

ω_R = (1/(R_Ecos φ))(∂v/∂λ) - (1/R_E) (∂u/∂φ)

= (1/(R_Ecos φ))(∂v/∂x) - (1/R_E) (dy/dφ)(∂u/∂y)

= (1/(R_Ecos φ))(∂v/∂x - ∂u/∂y)

= ω_M / (R_Ecos φ)

where ω_M is the voriticity in Mercator space (or simply ω as written in previous equations). The derivation of the equation above is indepedent of the choice of latitude and longitude, φ and λ, and so we can use our rotated coordinates, φ' and λ' - provided that we use the rotated Mercator coordinates to match.

This means that (R_E ω_R) can be calculated once u and v are read into the code and ω' can be calculated by simply multiplying this by cosφ'. If a grid point is given by φ_j and λ_i, then the voriticty at this grid point is given by

(R_E ω_R)_i,j = (v_i+1,j - v_i-1,j) / (cosλ_i (λ_i+1 - λ_i-1)) - (u_i,j+1 - u_i,j-1) / (φ_j+1 - φ_j-1)).

At boundaries, particularly the Poles, this equation becomes more complicated but this is discussed in the Adding extra points to pole equation. Having got (R_E ω_R) at grid points, it can be averaged onto any point in our domain in the same way that u and v are, and multiplied by cosφ' to find the vorticity in Mercator space.