Default horizontal diffusion coefficient in the reference system.

Aidan McDonald, IMS.

The fourth order implicit horizontal diffusion scheme described in HIRLAM technical report no. 17 has been installed in the reference system. The diffusion coefficient has been scaled to `1' in the namelist `namrun'. In this note is described the actual coefficient resulting from that choice and the thinking behind the scaling used.

Looking at Eq. (4.2) of the HIRLAM technical report no. 17, and putting Dy = Dx , the response function is

Writing k = 2p/L_x = 2p/(nDx) , R becomes

The e-folding time T^e is expressed as follows:

The assumption I have made is that if the grid is changed then the diffusion coefficient for the new grid will be such that e-folding time for the two-grid wave on the new grid will be the same as it was on the original grid. Putting T^e₁ = T^e₂ results in the following formula for the new diffusion coefficient:

In order to have the same e-folding time for two-grid wave on grid `1' (n<sub>1</sub> = 2) as for two-grid wave on grid `2' (n₂ = 2) we need

To get a reasonable estimate of K₁, notice that for 16K₂Dt₂ / Dx₂⁴ << 1,

The value of K₂ = 3.5×10¹⁴ for Dx₂ = 0.5^° and Dt₂ = 300s from which the others are scaled has been chosen as a result of testing the HIRLAM model with a particularly active data set; see HIRLAM newsletter no. 19, page 54.