Mass average

Let $F(x,y,\eta)$ be a variable defined on a 3D hybride grid. Consider a regular lon/lat grid cell with area $[\lambda_0,\lambda_1]\times[\phi_0,\phi_1]$ between halflevels $[\eta_{k-1/2},\eta_{k+1/2}]$. The field weighted with the mass is to be computed from:

$$\int\!\!\!\!\int\!\!\!\!\int F \mbox{d}m = \frac{1}{g} \int\!\!\!\!\int\!\!\!\!\int F \mbox{d}p \mbox{d}A ... ...\!\!\int\!\!\!\!\int F \frac{\mbox{d}p}{\mbox{d}\eta} \mbox{d}\eta \mbox{d}A$$ (3.37)

$$\approx \frac{1}{g} \int\!\!\!\!\int F_k \Delta p_k \mbox{d}A  =  \frac{1}{g} \int\!\!\!\!\int F_k (\Delta a_k + \Delta b_k p_s) \mbox{d}A$$ (3.38)

$$= \frac{1}{g} \int\!\!\!\!\int F_k (\Delta a_k + \Delta b_k \exp(\ln p_s)) \mbox{d}A$$ (3.39)

$$= \frac{1}{g} \int\!\!\!\!\int F_k (\Delta a_k + \Delta b_k \exp(\ln p_s)) \cos\phi \mbox{d}\phi \mbox{d}\lambda$$ (3.40)

The mass-average field is thus computed from:

$$\overline{F} =  \frac{\int\!\!\!\!\int F_k (\Delta a_k + ... ... b_k \exp(\ln p_s)) \cos\phi \mbox{d}\phi \mbox{d}\lambda}$$ (3.41)