Vertical Hybrid dry-hydrostatic mass Coordinate

The vertical coordinate is a so-called hybrid hydrostatic pressure coordinate. In ASP, it is defined as a function of the dry hydrostatic surface pressure, $\pi_{ds}$, and the dry atmospheric hydrostatic pressure, $\pi_d$:

$$\eta = f\left(\pi_d,\, \pi_{ds}\right)$$ (56)

It decreases monotonically from 1 at the surface (terrain) to 0 at the top of the modeled atmosphere, so that the boundary conditions are expressed as

$$\pi(\eta=1) =\pi_{ds}$$ (57)

$$\pi(\eta=0) =\pi_t \hskip.5in \left(p_t \geq 0\right)$$ (58)

where the constant dry hydrostatic pressure at the top of the model domain is defined as $\pi _t$.

The dry hydrostatic pressure at any location in the model can be expressed simply as a linear function of the surface pressure as

$$\pi_d = A_\eta + B_\eta \, \pi_{ds}$$ (59)

(as done in the ECMWF, ARPEGE, ECHAM, GFDL, etc... models). The $A_\eta$ and $B_\eta$ coefficients in Eq. 60 are a function of the vertical coordinate only, and thus only need be specified prior to model temporal integration (i.e. they remain fixed in time). They must respect the following constraints:

$$A_1 = 0, \,\, B_1=1$$ (60)

$$A_0 = \pi_t, \,\, B_0=0$$ (61)

The specific (dry) density can then be defined quite simply from Eq. 60 as

$$\mu_d = {\frac{\partial A_\eta}{\partial\eta}} + {\frac{\partial B_\eta}{\partial\eta}} \, \pi_{ds}$$ (62)

where the derivative terms need only be computed once as a pre-processing step.

The most common hybrid pressure coordinate is obtained by defining the $A_\eta$ and $B_\eta$ coefficients as

$$A_\eta =\pi_t(1-\eta), \,\, B_\eta=\eta$$ (63)

so that pressure is defined as

$$\pi_d = \pi_t(1-\eta) + \eta \pi_{ds}$$ (64)

This equation can then easily be solved for the vertical coordinate yielding

$$\eta = \sigma = {\frac {\pi_d - \pi_t }{\pi_{ds} - \pi_t}}$$ (65)

which is the classic so-called sigma coordinate. For example, the WRFv4 (and before) model uses this vertical coordinate (Skamarock et al., 2005). The specific density is obtained by taking the derivative of Eq. 65 with respect to $\eta$ which yields

$$\mu_d = \pi_{ds} - \pi_t$$ (66)

The specific density has no vertical dependence which simplifies a bit the system of dynamic equations. An example of the vertical pressure grid using Eq. 66 is shown in Fig. 1 for a 20-layer configuration. This was the original ASP grid configuration. As seen in Fig. 1, the pressure surfaces become progressively flatter away from the surface terrain.

Figure 1: A 20-layer sigma-coordinate model (Eq. 66) for an idealized bell-shaped mountain. The top pressure, $\pi _t$, is defined as 100 Hpa here, while the terrain-following surface pressure is indicated using the thick black line, $\pi _s$. The sea level pressure is 1013 Hpa. The highest vertical resolution has been defined to be in the lower atmosphere.

Note that, however, it has been shown that in upper levels of the atmosphere, horizontal pressure gradient errors can be reduced by forcing the coordinate surfaces to transform to pressure surfaces in levels as low as 400 mb (e.g. Z. Janjic and DiMego, 2004). In order to introduce this functionality, we redefine the $A$ and $B$ coefficients in a more general manner using

$$A_\eta = \pi_m \left( 1-\sigma^\prime \right)\delta_{\sigma^\prime} \,+\, ... ...ight) \left[ \pi_m \left( 1+\sigma^\prime \right) - \sigma^\prime \pi_t \right]$$ (67)

$$B_\eta = \sigma^\prime \delta_{\sigma^\prime}$$ (68)

where the modified sigma, $\sigma^\prime$, is defined over the range

$$\left( -1 \leq \sigma^\prime \leq 1 \right)$$ (69)

where $\sigma^\prime$ is just a dummy variable which is not actually used in the model, but only in the computations of the vertical grid during pre-processing. The delta function in Eq.s 68-69 is defined as

$$\delta_{\sigma^\prime} = 1 \,\,\, \left( \sigma^\prime \geq 0 \right)$$ (70)

$$\delta_{\sigma^\prime} = 0 \,\,\, \left( \sigma^\prime < 0 \right)$$ (71)

The relationships between $\sigma^\prime$ and pressure for each of the two aforementioned regions are then defined as

$$\sigma^\prime = {\frac{\pi_d - \pi_m }{ \pi_m - \pi_t}} \,\,\, \left( \pi_d < \pi_m \,:\, 0 \geq \sigma^\prime \geq -1 \right)$$ (72)

$$\sigma^\prime = {\frac{\pi_d - \pi_m }{ \pi_{ds} - \pi_m}} \,\,\, \left( \pi_d \geq \pi_m \,:\, 1 \geq \sigma^\prime \geq 0 \right)$$ (73)

Once the new $A_\eta$ and $B_\eta$ coefficients have been computed from Eq.s 68-69, the actual $\eta$ vertical coordinate can be computed from

$$\eta = \frac{A_\eta + B_\eta p_0 - \pi_t}{p_0 - \pi_t }$$ (74)

where $p_0$ is the standard sea level pressure. The transition (to flat surfaces) pressure is at $\pi_m$. Below this pressure value a terrain following coordinate is used, and a pure pressure coordinate is used above. Note that $\mu_d$ is constant and space and time above the transition pressure: it is simply defined as $\mu_d=\pi_m - \pi_t$. Below the transition pressure, it is defined as $\mu_d=\pi_{ds} - \pi_m$. An example of the resulting pressure grid (similar to the current grid used in WRF-NMM) is shown in Fig. 2.1.8 for a 20-layer model configuration.

Figure 2: A 20-layer hybrid-pressure coordinate model (Eq. 75) for an idealized bell-shaped mountain. The coordinate surfaces are indicated using the thin black lines. The top pressure is defined as $p_t$=100 Hpa. The transition from terrain-following to pressure surfaces occurs at $p_m$=360 Hpa. The surface (terrain following) hydrostatic pressure is indicated by $p_s$. The sea level pressure is 1013 Hpa. The highest vertical resolution has been defined to be in the lower atmosphere.

Fig. 2.1.8

As a final note, we can how define the slope of the pressure surfaces from Eq. 60 as simply

$${\frac{\partial \pi_d}{\partial \pi_{ds}}} = B_\eta$$ (75)

so that for constant (flat) pressure surfaces, such as those in the upper part of the atmosphere (i.e. above $\pi_m$), then $\partial \pi_d/\partial \pi_{ds}=0$.