FV3DYCORE  Version 2.0.0
Variables and notation

(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
Qi (*) Density of tracer i (also written Q as a generic tracer density)
qi Mixing ratio of tracer i, defined with respect to total air mass; equal to Qi/δp*
qv 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
ΔAc, Δxc, Δyc Dual-grid cell areas and cell face lengths
Δt Lagrangian dynamics (or acoustic) time step
ΔT Vertical remapping interval
Δτ Physics time step
cpd Specific heat of dry air at constant pressure
Rd Gas constant for dry air
cp Variable specific heat of moist air
κ R/cp, 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 δx 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 δx a 1/Δx 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 Θv :

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

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

where ε = 1 + Rv/Rd and κ = R/cp. Note that we do not include the arbitrary constant factor p0κ in our definition of potential temperature; our form is the equivalent to setting p0 = 1Pa but simpler and more computationally efficient.