Vertical coordinate system

The ECMWF model uses a general, terrain-following, vertical coordinate $\eta$. The definition of $\eta$ leads to a specification of model levels at hybride pressure and $\sigma$ coordinates. The same model levels are addopted for TM#, although for stratospheric studies some levels in the troposhere are sometimes skipped.

Coordinate $\eta$ satisfies two conditions [Simmons and Burridge, 1981]:

$$\eta = h(p,p_s)$$ (3.29)

$$h(0,p_s) = 0\qquad,\qquad h(p_s,p_s) = 1$$ (3.30)

The exact definition of $h$ is never used. Instead, the vertical discretization evaluates entities at full levels $\eta_k$, $k=1,\dots,NLEV$, and intermediate half levels $\eta_{1/2},\eta_{1+1/2},\dots$. The pressure at the the half levels is defined by hybride coefficients:

$$p_{k+1/2} = a_{k+1/2} +  b_{k+1/2} p_s \qquad \mbox{[Pa]}$$ (3.31)

Pressure differences are defined through:

$$\Delta p_k = p_{k+1/2} -  p_{k-1/2} =  \Delta a_k +  \Delta b_k p_s$$ (3.32)

$$\Delta a_k = a_{k+1/2} -  a_{k-1/2}$$ (3.33)

$$\Delta b_k = b_{k+1/2} -  b_{k-1/2}$$ (3.34)

The hybride coefficients provide an implicit definition of $\eta$ at half levels [White, 2000, eq. 3.8]; this definition is however not necessary for implementation:

$$\eta_{k+1/2} =  a_{k+1/2}/p_o +  b_{k+1/2} \quad,\quad p_o = 1013.25 \mbox{Pa}$$ (3.35)

According to [Simmons and Burridge, 1981] and [White, 2000], an integral over $\eta$ between two half levels of a function $F\partial p/\partial \eta$, with $F$ defined at the full level, is approximated by assumption of constant $F(\eta)=F_k$ and linear interpolated pressure $p(\eta)$:

$$\int\limits_{\eta_{k-1/2}}^{\eta_{k+1/2}} F\frac{\partial p... ... \Delta p_k  =  F_k \Delta a_k +  F_k p_s \Delta b_k$$ (3.36)