Final Flux Form prognostic Equations

Using Eq. 28, the prognostic equations can be expressed using the dry hydrostatic mass in flux form as

$${\frac{\partial U}{\partial t}} + \nabla \cdot {\bf V} U + {\frac{\partial {\dot\eta}U}{\partial\eta}} = - (1+\epsilon) \mu_d {\frac{\partial \Phi}{\partial x}} \,-\, \mu... ...tial\Pi}{\partial x}} + \mu_d \,f_{u,crv} + \mu_d\left(D_u + {\dot{S}}_u\right)$$ (30)

$${\frac{\partial V}{\partial t}} + \nabla \cdot {\bf V} V + {\frac{\partial {\dot\eta}V}{\partial\eta}} = - (1+\epsilon) \mu_d{\frac{\partial \Phi}{\partial y}} \,-\, \mu_... ...al\Pi}{\partial y}} + \mu_d \,f_{v,crv} + \mu_d \left(D_v + {\dot{S}}_v \right)$$ (31)

$${\frac{\partial W}{\partial t}} + \nabla \cdot {\bf V} W + {\frac{\partial {\dot\eta}W}{\partial\eta}} = g \, \mu_d \, \epsilon + \mu_d \,f_{w,crv} + \mu_d \,D_w$$ (32)

$${\frac{\partial \Theta_\rho}{\partial t} } + \nabla \cdot {\bf V} \Theta_\rho + {\frac{\partial {\dot\eta}\Theta_\rho}{\partial\eta}} = \mu_d \left( D_{\theta_\rho}+ {\dot{S}}_{\theta_\rho} \right)$$ (33)

$${\frac{\partial \pi_{d,s}}{\partial t}} = - \int_0^1 \left( \nabla \cdot {\bf V} \mu_d \right) \,d\eta$$ (34)

$${\frac{\partial\Phi}{\partial t}} + {\bf V} \cdot\nabla \Phi + {\dot\eta} {\frac{\partial\Phi}{\partial\eta}} = g \,w$$ (35)

where $\pi_{d,s}$ represents the dry surface hydrostatic pressure, and the mass-variables are defined as $U=u\,\mu_d$, $V=v\,\mu_d$, $W=w\,\mu_d$, $\Theta=\theta\,\mu_d$. The $D$ terms in the above equations represent horizontal and vertical (numerical and physically-based) diffusion. Any relaxation terms (such as for certain variables in a so-called upper sponge layer or lateral boundaries in LAM mode, are also included herein). In addition, divergence damping is included in $D_u$ and $D_v$. The ${\dot{S}}$ terms represent additional diabatic process tendencies (from deep convection, radiation, etc.) and they are detailed in the physics description of ASP. The surface hydrostatic pressure tendency equation, Eq. 34, results from the vertical integration of Eq. 28.

In order to close the system of equations, the following diagnostic relationships have been used

$$p = p_0 {\left( {\frac{R_d\,\theta_\rho} {p_0\,\alpha_d\,\varphi_q}} \right)}^{1/(1-\varsigma)}$$ (36)

$$\frac{\partial\Phi}{\partial{\Pi_\pi}_d} = -\alpha_d\,\vartheta_{\pi\,d}$$ (37)

$$\vartheta_{\pi\,d} = p_0^{\varsigma} \varsigma^{-1} \pi_d^{1-\varsigma}$$ (38)

$$\epsilon = \varphi_q \frac{\partial p}{\partial\pi_d}-1$$ (39)

Note that Eq. 37 comes from Eq. 25. Since $\varsigma=R_d/C_{pd}$, we can write ${\varsigma}/({1-\varsigma})=R_d/C_{vd}$ where the specific heat capacity of dry air at constant volume is defined as $C_{vd} = C_{pd}-R_d$. Note that $\vartheta_\pi=\partial\pi/\partial\Pi_\pi$, which applies to both moist and dry hydrostatic pressure. Eq. 40 is defined from Eq. 10 using the relation between dry and moist hydrostatic mass defined in Eq. 29.

The dry hydrostatic Exner function defined as

$${\Pi_\pi}_d = {\left({\frac{\pi_d}{p_0}}\right)}^\varsigma$$ (40)

while the Exner function (including water vapor effects) is computed from

$$\Pi = {\left({\frac{p}{p_0}}\right)}^\varsigma$$ (41)

We can define several general relationships (some of which are repeated here for completeness) between the moist and dry variables:

$$\alpha = \alpha_d \,\varphi_q$$ (42)

$$q_j = r_j \,\varphi_q$$ (43)

$$q_j \,\mu = r_j \,\mu_d$$ (44)

$$q_j \,\rho = r_j \,\rho_d$$ (45)

Note that if physics schemes use the specific humidity, $q_j$, then any mass weighted vertical integrations must use the moist hydrostatic mass, $\mu$, from Eq. 45 in order to conserve mass (i.e. $r_j,\,\mu_d$) consistent with the mixing ratio flux form prognostic equations. From Eq.43 and Eq.29, the geopotentials are the same for the moist or the dry equations:

$${\frac{\partial\Phi}{\partial\eta}} = -\alpha_d \,\mu_d = -\alpha\,\mu$$ (46)

Note that as the pressure approaches the hydrostatic pressure, $\partial p/\partial\pi \rightarrow 1$. The non-hydrostatic part of the model is an add-on module, and for large scales turning it off (i.e. $\epsilon=0$) produces nearly identical results to when it is on (at more coarse grid resolutions) and saving CPUs. This module consists in evaluating the $w$ and $\Phi$ prognostic equations thus when the non-hydrostatic option is off, Eq.s 32 and 35 (and the associated forcing functions) are not evaluated, $\Pi=\Pi_\pi$ (so that $\epsilon=0$). Also, $\alpha=R T_\rho/\pi$ in this case. Map factors are taken into account in the usual manner, along with full curvature terms in the horizontal momentum equations and this is detailed in Section 2.11.

The generic scalar equation (water variables, passive tracers, etc...) is given as

$${\frac{\partial \varphi\mu_d }{\partial t} } + \nabla \cdot {\bf ... ...dot\eta}\mu_d\varphi }{\partial\eta}} = \mu_d\,{\dot\varphi} + \mu_d\,D_\varphi$$ (47)

where $\varphi$ is some scalar and ${\dot\varphi}$ is the scalar sink term (owing to microphysical, turbulent, convective or surface processes). For the dry mass form equations, mixing ratio, $r_j$, is used for the scalars, and ASP also predicts the turbulent kinetic energy (TKE), represented by $e$, using

$${\frac{\partial r_j \mu_d }{\partial t} } + \nabla \cdot {\bf V} \, \mu_d \,r_j + {\frac{\partial {\dot\eta}\mu_d \, r_j }{\partial\eta}} = \mu_d\,{\dot r}_j + \mu_d\,D_{r_j} \hskip.5in \left( r_j = r_v,\,r_{cl},\,r_{ci},\,r_{pl},\,r_{pl} \right)$$ (48)

$${\frac{\partial e \mu_d }{\partial t} } + \nabla \cdot {\bf V} \, \mu_d \,e + {\frac{\partial {\dot\eta}\mu_d \, e }{\partial\eta}} = \mu_d\,{\dot e} + \mu_d\,D_e$$ (49)

where $r_v,\,r_{cl},\,r_{ci},\,r_{pl},\,r_{pl}$ represent the mixing ratios for water vapor, cloud liquid water, cloud ice water, liquid precipitation and solid precipitation, respectively.

The prognostic equations (Eq.s 30-34) are those which are integrated in time in the dynamic core in the ASP model. Eq.s 49-50 are integrated outside of the dynamic core but using the dynamic tendencies within the RK time step loop (to be described later).