Governing Equations

The prognostic equations consisting in two horizontal momentum, hydrostatic, heat, water vapor and continuity equations are expressed in Cartesian coordinates, respectively, as

$${\frac{\partial u}{\partial t}} + {\bf V} \cdot\nabla u + {\dot\eta} {\frac{\partial u}{\partial\eta}} = f \,v - {\frac{\partial \Phi}{\partial x}} - \alpha{\frac{\partial p}{\partial x}} + D_u$$ (1)

$${\frac{\partial v}{\partial t}} + {\bf V} \cdot\nabla v + {\dot\eta} {\frac{\partial v}{\partial\eta}} = - f \,u - {\frac{\partial \Phi}{\partial y}} - \alpha{\frac{\partial p}{\partial y}} + D_v$$ (2)

$${\frac{\partial\Phi}{\partial\eta}} = -\alpha {\frac{\partial p}{\partial\eta}}$$ (3)

$${\frac{\partial T }{\partial t}} + {\bf V} \cdot\nabla T + {\dot\eta} {\frac{\partial T }{\partial\eta}} = {\frac{\alpha\omega}{C_p}} + {\frac{\dot Q}{ C_p}} + D_T$$ (4)

$${\frac{\partial q }{\partial t} } + {\bf V} \cdot\nabla q + {\dot\eta} {\frac{\partial q }{\partial\eta}} = {S_q} + {D_q}$$ (5)

$${\frac{\partial}{\partial\eta}}\left({\frac{\partial p}{\partial ... ...tial}{\partial\eta}} \left({\dot\eta}{\frac{\partial p}{\partial\eta}}\right) =$$ 0 (6)

where a generalized hybrid coordinate (see Kasahara (1974)) is used (indicated here using $\eta$ ). The $D_\phi$ terms in the above equations include divergence damping, horizontal and vertical diffusion. ${\dot Q}$ represents diabatic heating from convection, radiative transfer, latent heating (related to large scale precipitation), and $S_q$ represents the water vapor source-sink term. All of these source-sink terms will be described in subsequent sections. The ideal gas law is defined as

$$\alpha = {\frac{1}{\rho}} = {\frac{R_d\,T_v}{p}}$$ (7)

where the virtual temperature is expressed as

$$T_v = T \left[1 + \left({\frac{R_v}{R_d}}-1\right)q \right]$$ (8)

where $R_d$ and $R_v$ represent the gas constants for dry air and water vapor, respectively. Finally, the heat capacity of the air at constant pressure is defined as

$$C_p = C_{pd} \left(1+\epsilon_q q\right)$$ (9)

where

$$\epsilon_q = \left({C_{pv}/C_{pd}}\right)-1$$ (10)

and $C_{pd}$ and $C_{pv}$ represent the specific heats for dry air and water vapor at constant pressure, respectively.

The hydrostatic equation can be expressed in several forms:

$${\frac{\partial\Phi}{\partial\eta}} = -{\frac{R_d T_v }{p}}{\frac... ...al\eta}} = - \alpha\mu = - \theta_v \,C_{pd} {\frac{\partial\Pi}{\partial\eta}}$$ (11)

where the following definitions are used:

$${\cal P} = {\rm ln}(p)$$ (12)

$$\mu = {\frac{\partial p}{\partial\eta}}$$ (13)

$$\theta_v = {\frac{T_v}{\Pi}}$$ (14)

$$\Pi = {\left({\frac{p}{p_0}}\right)}^{R/C_{pd}}$$ (15)

where $\mu$ represents the specific density, $\theta_v$ is the virtual potential temperature, and $\Pi$ represents the Exner function ($p_0=1000$ HPa). Note that the effect of moisture is included in Eq. 4 via use of the virtual temperature.

Finally, using the identity

$$\nabla p = {\frac{\partial p}{\partial p_s}} \nabla p_s$$ (16)

we can rewrite the horizontal momentum equations with the modified pressure gradient terms as

$${\frac{\partial u}{\partial t}} + {\bf V} \cdot\nabla u + {\dot\eta} {\frac{\partial u}{\partial\eta}} = f \,v - {\frac{\partial \Phi}{\partial x}} - \alpha {\frac{\partial p}{\partial p_s}} {\frac{\partial p_s}{\partial x}} + D_u$$ (17)

$${\frac{\partial v}{\partial t}} + {\bf V} \cdot\nabla v + {\dot\eta} {\frac{\partial v}{\partial\eta}} = - f \,u - {\frac{\partial \Phi}{\partial y}} - \alpha {\frac{\partial p}{\partial p_s}} {\frac{\partial p_s}{\partial y}} + D_v$$ (18)

The advantage of the pressure gradient terms in the above form is that the horizontal gradient of pressure need only be evaluated at the surface. The $\partial p/\partial p_s$ term can be computed once the vertical coordinate has been specified.