Variable transformations

We now formulate the final form for the closed set of equations which will be integrated in time. First, we choose to rewrite several of the thermodynamic terms using the Exner function which is defined as

$$\Pi = {\left({\frac{p}{p_0}}\right)}^\varsigma$$ (11)

where $\varsigma=R_d/C_{pd}$ and the heat capacity of dry air at constant pressure is defined as $C_{pd}$ and $p_0=10^5$ Pa. The Exner function and the temperature are related using the definition of the potential temperature:

$$\theta = \frac{T}{\Pi}$$ (12)

By using the potential temperature, $\theta$, as opposed to temperature, $T$ in the thermodynamic equation, Eq. 4 is rewritten as

$$\frac{d\theta}{dt} = \frac{{\dot{S}}_T}{\Pi} = {\dot{\theta}}$$ (13)

so that the so called alpha-omega term in Eq. 4 (involving $\omega$) vanishes.

In many numerical weather forecasting systems, the pressure gradient terms in the horizontal momentum equations are recast using either the natural log of pressure or the Exner function. One of the main reasons is that the geopotential estimates using the hydrostatic equation are improved in the upper atmosphere using either of these two forms compared to simply using pressure. In order to be consistent with using the potential temperature as the thermodynamic prognostic variable, we use the Exner function (Eq. 11) to cast the horizontal pressure gradient (in Eq.s 1-2) as

$$\alpha \, \nabla p = \frac{\alpha \,p\,C_{pd}}{\Pi \,R_d} \nabla\... ...\alpha \,p\,\theta \,C_{pd}}{T\,R_d} \nabla\Pi \,=\,\theta \,C_{pd}\, \nabla\Pi$$ (14)

For a consistent finite difference form of the pressure gradient terms, we rewrite the hydrostatic equation (Eq. 8) in terms of the hydrostatic Exner function, ${\Pi_\pi}$, defined using Eq. 11 as

$${\Pi_\pi} = {\left({\frac{\pi}{p_0}}\right)}^\varsigma$$ (15)

so that

$$\frac{\partial\Phi}{\partial\pi} \,=\, \frac{\partial {\Pi_\pi}}{... ...Pi_\pi}}\,=\, \frac{1}{\vartheta_{\pi}} \frac{\partial\Phi}{\partial {\Pi_\pi}}$$ (16)

where we have defined

$$\vartheta_{\pi} = \frac{\partial\pi}{\partial{\Pi_\pi}} \,=\, \varsigma^{-1} \frac{\pi}{\Pi_\pi}$$ (17)

Alternatively, $\vartheta_{\pi}$ can also be expressed as only a function of $\pi$

$$\vartheta_{\pi} = \frac{C_{pd}\,p_0^{R_d/C_{pd}}}{R_d} \pi^{C_{vd}/C_{cp}}\,=\, \frac{p_0^{\varsigma}}{\varsigma} \pi^{1-\varsigma}$$ (18)

or as only a function of $\Pi_\pi$

$$\vartheta_{\pi} = \varsigma\,p_0\, {\Pi_\pi}^{(1-\varsigma)/\varsigma}$$ (19)

and obviously the same relationships hold for $p$ and $\Pi$. The hydrostatic equation can be expressed as a function of the inverse air density as

$$\frac{\partial\Phi}{\partial\eta} = - \alpha \frac{\pi}{\Pi_\pi} ... ...partial\eta} = - \alpha\,\vartheta_{\pi} \frac{\partial{\Pi_\pi}}{\partial\eta}$$ (20)

Note that the last form on the RHS of Eq. 20 is very close to another commonly used form of the hydrostatic equation (not used herein):

$$\frac{\partial\Phi}{\partial\eta} = - \alpha\,\pi \frac{\partial {\rm ln}(\pi)}{\partial\eta}$$ (21)

and the two forms, Eq. 20 and Eq. 21 give very similar results for geopotential given the same hydrostatic pressure and density distribution (especially in the upper atmosphere). Thus to evaluate the geopotential in Eq. 20, one must multiply the inverse density by a factor which only depends on the hydrostatic pressure, $\pi$. It turns out that this form is useful for the implicit solution of $w$ and $p$ (to be discussed in the numerical solution section). As grid size increases to about 10 km then $p \rightarrow \pi$ ( $\Pi \rightarrow {\Pi_\pi}$).