Adding extra points to poles

Extra latitude levels beyond poles

For a given location, four grid points are used to calculate a weighted average for u, v and the vorticity at this location. It is simple to find these points when the location is within the extremities of the latitude-longitude grid for the wind vectors. However, it's helpful to introduce extra grid points beyond these extremities for when the location falls outside the extremities of the latitude-longitude grid.

For example, if the resolution in the longitude direction is 0.5°, then it's likely to run from 0° to 355.5°. It's possible to get flight paths greater than 355.5° but less than 360.0°, and so we introduce an imaginary longitude level at 360.0 °, which has the same velocities as the longitude level at 0.0°, as 360.0° is the same location as 0°.

The same issue appears at the pole, since the highest and lowest latitude levels will typically not reach 90° and -90° - but flight paths can reach these latitudes. In the case of the longitude levels, the lowest longitude level is already available for adding to the end of the highest longitude levels. This is almost true at the highest latitude level, because it is possible to add a latitude level north of this, which is velocities from this level but each point is 180° further East (or West), and of opposite sign. If a longitude level doesn't exist at 180° from each point, a weighted average of the nearest longitude levels can be used. Say the latitude resolution is 0.5° and the highest latitude is at 89.5°, this imaginary level will be at 90.5°. However, the calculation of vorticity assumes the gaps between latitude levels are approximately the same, and so for this calculation it's helpful to pick an imaginary latitude level with the similar gap from the highest level as the gaps found between the real latitude levels - and linearly interpolate onto this level. The same is true for an imaginary level near the South Pole.

Let the latitude levels, lat_i run from i = 1 to N, and we'll introduce imaginary level near the south pole at i=0 and near the north pole at i=N+1, so that

lat₀ = 2 lat₁ - lat₂

lat_N+1 = 2 lat_N - lat_N-1

and let value of u at 180° east/west be represented by u_opp,1 and u_opp,N for the most southern and northern latitude levels respectively. Hence

u_N+1 ≈ u_N + (du/dlat)|_N (lat_N+1 - lat_N).

and -u_opp,N is our u at (180 - lat_N)° and so (du/dlat)|_N ≈ ((-u_opp,N) - u_N) / ( 2(90 - lat_N)). Thus

u_N+1 ≈ u_N - 0.5 & (u_opp,N + u_N) (lat_N+1 - lat_N) / (90 - lat_N).

And simiarly,

u₀ ≈ u₁ + 0.5 & (u_opp,1 + u₁) (lat₁ - lat₀) / (lat₁ + 90.0).

v at lat₀ and lat_N+1 can be calculated in the same way.

Calculating vorticity at poles

As well as u and v, we require the vorticity, ω, at lat₀ and lat₁. The directions north and south have no unique meaning at the poles, which is a barrier to calculating u and v at the poles. However, ω does not rely on a direction and so can be calculated at the poles. For a given longitude, λ, the vorticity at the north pole should be given by

ω_North = (∂v/∂x) - (∂u/∂y)

= (u_N(λ+90) - (-u_N(λ+270))) / ( 2R_E(90 - lat_N) )- ((-u_N(λ+180)) - u_N(λ)) / ( 2R_E(90 - lat_N) )

= (u_N(λ) + u_N(λ+90) + u_N(λ+180) + u_N(λ+270)) / ( 2R_E(90 - lat_N) )

where R_E is the radius of the Earth. This should be the same whatever the choice of λ. And so it's sensible to take an average value by using all the u_N values available, and by summing λ from 0° to 90°, so that

ω_North = (2 / (N R_E(90 - lat_N))) Σ^N/4_i=1 (u_N,λ + u_N,λ+90 + u_N,λ+180 + u_N,λ+270)

= (2 / (N R_E(90 - lat_N))) Σ^N_i=1 u_N,i

and it's a similar equation to calculate ω_south. Calculating ω₀ and ω_N+1 is then similar to the derivation of u_N except (dω/dlat) can be calculated with the omega values at the poles, so that

ω₀ = ω₁ - (ω₁ - ω_south) (lat₁ - lat₀) / (lat₁ + 90)

ω_N+1 = ω_N + (ω_north - ω_N) (lat_N+1 - lat_N) / (90 - lat_N).