Governing Equations

The fully compressible governing set of prognostic equations for the horizontal momentum components, the vertical momentum, thermodynamic, hydrostatic mass continuity, geopotential, respectively, for a dry atmosphere are expressed in Cartesian coordinates and the normalized mass-pressure vertical coordinate following Lapise (1992) are expressed as

$${\frac{\partial u}{\partial t}} + {\bf V} \cdot\nabla u + {\dot\eta} {\frac{\partial u}{\partial\eta}} = -(1+\epsilon) {\frac{\partial \Phi}{\partial x}} + {\frac{\partial \Phi}{\partial\pi}} {\frac{\partial p}{\partial x}} + f_{u,crv}$$ (1)

$${\frac{\partial v}{\partial t}} + {\bf V} \cdot\nabla v + {\dot\eta} {\frac{\partial v}{\partial\eta}} = -(1+\epsilon) {\frac{\partial \Phi}{\partial y}} + {\frac{\partial \Phi}{\partial\pi}} {\frac{\partial p}{\partial y}} + f_{v,crv}$$ (2)

$${\frac{\partial w}{\partial t}} + {\bf V} \cdot\nabla w + {\dot\eta} {\frac{\partial w}{\partial\eta}} = g\,\epsilon + f_{w,crv}$$ (3)

$${\frac{\partial T}{\partial t}} + {\bf V} \cdot\nabla T + {\dot\eta} {\frac{\partial T}{\partial\eta}} = {\frac{\alpha\omega}{C_{p}}} + {\dot{S}}_T$$ (4)

$${\frac{\partial\Phi}{\partial t}} + {\bf V} \cdot\nabla \Phi + {\dot\eta} {\frac{\partial\Phi}{\partial\eta}} = g \,w$$ (5)

$${\frac{\partial\mu}{\partial t}}+ \nabla \cdot \left({\bf V} \mu\right) + {\frac{\partial{\dot\eta}\mu}{\partial\eta}} =$$ 0 (6)

together with the equation of state, the diagnostic relationship for the inverse density (or hydrostatic equation) and the (hydrostatic) specific density, respectively:

$$p = {\frac{R_d\,T}{\alpha}}$$ (7)

$${\frac{\partial\Phi}{\partial\pi}} = -\alpha$$ (8)

$$\mu = {\frac{\partial\pi}{\partial\eta}}$$ (9)

$$\epsilon = \frac{\partial p}{\partial\pi}-1$$ (10)

A normalized hybrid mass coordinate, $\eta$ (see Section 2.1.8) is used. The hydrostatic pressure is represented by $\pi$, and the non-hydrostatic pressure is given by $p$. The parameter $\epsilon$ represents the vertical acceleration and controls the non-hydrostatic dynamics: it is generally small to negligible (i.e. $p \rightarrow \pi$) for grid resolutions above approximately 10 km. The curvature terms in the momentum equations are represented by $f_{crv}$ and their form depends on the map projection (and will be presented in a subsequent section): ASP supports Mercator, Lambert and Platte-Carré grids. Eq. 5 is simply the definition of $w$ in the mass coordinate system. Note that the mass-continuity equation in mass-coordinates is Eq. 6. Finally, ${\dot{S}}_T$ represents a diabatic heating source/sink.