Perturbation about a reference state

In order to improve the numerical accuracy for both the horizontal (in the $u$ and $v$ prognostic equations) and vertical pressure gradient (in the $w$ prognostic equation) terms, a linearization about a hydrostatic reference state is often used. A simple linear perturbation method is used for the variables

$$\Phi = \Phi^\ast \,+\, \Phi^\prime$$ (180)

$$\theta_\rho = \theta^\ast \,+\, \theta_\rho^\prime$$ (181)

$$\pi_d = \pi_d^\ast \,+\, \pi_d^\prime$$ (182)

$$p = p^\ast \,+\, p^\prime$$ (183)

$$\Pi = \Pi_{\pi d}^\ast \,+\, \Pi^\prime$$ (184)

where $\ast$ indicates the hydrostatically balanced background state and $\prime$ indicates a perturbation about this state. The horizontal pressure gradient from the flux-form equations is written as

$$PG_{xy} = \mu_d \left( \nabla\Phi + C_{pd}\theta_\rho \nabla\Pi\right) + \mu_d\epsilon\nabla\Phi$$ (185)

where using Eq. 40 we define

$$\zeta = \mu_d\epsilon = \varphi_q \frac{\partial p}{\partial\eta} - \frac{\partial\pi_d}{\partial\eta}$$ (186)

Plugging Eq.s 181-185 and Eq. 187 into Eq. 186 we have

$$PG_{xy} = \mu_d \left( C_{pd}\theta^\ast \nabla \Pi_{\pi d}^\ast \,+\, C_{pd}\the... ...ime\nabla\Pi^\prime \,+\, \nabla\Phi^\ast \,+\, \nabla\Phi^\prime \right) \,+\,$$ (187)

$$\zeta \nabla\Phi$$ (188)

Several terms cancel out of Eq.193. First, the background hydrostatic terms cancel out using the hydrostatic equation:

$$C_{pd}\theta^\ast \nabla \Pi_{\pi d}^\ast = -\nabla\Phi^\ast$$ (189)

Next, substituting the perturbation relationships for $p$ and $\pi_d$ into the vertical pressure gradient difference term, Eq. 187, yields

$$\zeta =\varphi_q \left( \frac{\partial p^\ast}{\partial\eta} + \f... ...ac{\partial\pi_d^\ast}{\partial\eta} -\frac{\partial\pi_d^\prime}{\partial\eta}$$ (190)

The background state for both the hydrostatic and non-hydrostatic pressures are assumed to be the same (i.e. $p^\ast=\pi_d^\ast$), thus Eq. 191 can be rewritten as

$$\zeta = \left(\varphi_q -1\right) \frac{\partial\pi_d^\ast}{\part... ...ac{\partial p^\prime}{\partial\eta} - \frac{\partial\pi_d^\prime}{\partial\eta}$$ (191)

Thus Eq. 193 is expressed as

$$PG_{xy} = \mu_d \left( C_{pd}\theta_\rho^\prime \nabla \Pi_{\pi d}^\ast\,... ...o^\prime\nabla\Pi^\prime \,+\, \nabla\Phi^\prime \right) \,+\, \zeta \nabla\Phi$$ (192)

$$=\mu_d \left( C_{pd}\theta_\rho^\prime \nabla \Pi_{\pi d}^\ast\,+... ...ial p^\prime}{\partial\eta} - \frac{\partial\pi_d^\prime}{\partial\eta} \right]$$ (193)

$$=\mu_d \Bigg\lbrace C_{pd}\theta_\rho^\prime \nabla \Pi_{\pi d}^\... ...partial\pi_d} - \frac{\partial\pi_d^\prime}{\partial\pi_d} \right] \Bigg\rbrace$$ (194)

The vertically varying hydrostatic basic state values are defined using a dry time constant hydrostatic background state. Note that the form in Eq. 194 is used in the WRF version 4 model.

Note that unlike the Simmons and Burridge (1981) method used for the omega core (with log-pressure) which necessitates the use of a perturbation form of the $PG$, numerical tests have shown that a perturbation form is not needed for the Exner-based core (as formulated herein) so for this core it is an option. But due to the added numerical accuracy, currently the perturbation form is the default used by ASP. The vertical pressure gradient term in the $w$ equation, $PG_z=\epsilon\mu_d=\zeta$, is defined in perturbation form in Eq. 192.