(This information is a reproduction of Appendix B excluding B.2 and B.3 from LPH17)

Variables	Notation
`u,v`	(*) D-grid winds
`w`	(*) Explicit vertical velocity
δ p^*	(*) Layer hydrostatic pressure thickness, proportional to mass
Θ _v	(*) Virtual potential temperature
δ _z	(*) Layer geometric depth
ρ	Total (air and all water species) mass density, equal to -δp^*/gδz
Q_i	(*) Density of tracer i (also written Q as a generic tracer density)
q_i	Mixing ratio of tracer i, defined with respect to total air mass; equal to Q_i/δp^*
q_v	Mixing ratio of water vapor
p	Total cell-mean pressure
p^*	Cell-mean hydrostatic pressure
p'	Cell-mean nonhydrostatic pressure component, equal to p - p^*
ΔA, Δx, Δy	D-grid cell areas and cell face lengths
ΔA_c, Δx_c, Δy_c	Dual-grid cell areas and cell face lengths
Δt	Lagrangian dynamics (or acoustic) time step
ΔT	Vertical remapping interval
Δτ	Physics time step
c_pd	Specific heat of dry air at constant pressure
R_d	Gas constant for dry air
c_p	Variable specific heat of moist air
κ	R/c_p, where R is the (variable) gas constant of moist air
i, j, k	Spatial grid-cell indices for the local x-, y-, and z-directions, respectively, given as subscripts
n	time index
m	tracer index, given as a subscript
n	time index, given as a superscript: tⁿ = nΔt

Here, a (*) indicates a prognostic variable. All variables are cell-means (or face-means for horizontal winds) unless otherwise noted. The differencing notation used in this document follows that of LR96, LR97, and L04, in which the operator δ_x φ is defined as a centered-difference operator:

\[ (\delta_x \phi)_{i+1/2} = \phi_{i+1} - \phi_i \tag {B.1} \]

The indices on dependent variables are suppressed unless explicitly needed.) This definition differs from the similar operators of in the literature intended to be second-order discretizations of a derivative; to do this with our definition of δ_x a 1/Δ_x term would be needed to complete the discrete derivative.

B.1 Important Relations

Cell-mean pressure:

\[ p^* = \frac{\delta p^*}{\delta \log p^*} \; \mathrm{(hydrostatic)} \\. \tag {B.2} \]

\[ p = \rho R_d T_v = - \frac{R_d}{g} \frac{\delta p^* T_v}{\delta z} \mathrm{(nonhydrostatic)} \tag {B.3} \]

Conversion from temperature T to virtual potential temperature Θ_v :

\[ T_v = T \left ( 1 + \epsilon q_v \right ) \tag {B.4} \]

\[ \Theta_v = T_v / p^\kappa \tag {B.5} \]

where ε = 1 + R_v/R_d and κ = R/c_p. Note that we do not include the arbitrary constant factor p₀^κ in our definition of potential temperature; our form is the equivalent to setting p₀ = 1Pa but simpler and more computationally efficient.