Notation
This appendix establishes a common notation across the documentation and source. Each entry lists a mathematical symbol, the Unicode form commonly used in the codebase, and a brief description. Mathematical symbols are shown with inline math, while the Unicode column shows the exact glyphs used in code.
| mathematical symbol | unicode | description |
|---|---|---|
| $\rho$ | ρ | Density |
| $\boldsymbol{u} = (u,v,w)$ | u, v, w | Velocity components (x, y, z) |
| $ρ \, u,\; ρ \,v,\; ρ \, w$ | ρu, ρv, ρw | Momentum components |
| $T$ | T | Temperature |
| $\alpha$ | α | Specific volume, $α = 1/ρ$ or $α = R^{m} T / p_r$ in anelastic |
| $ρ e$ | ρe | Moist static energy density (conservative variable) |
| $e$ | e | Moist static energy per unit mass, $e = c^{pd} \, θ$ |
| $ρ q^{t}$ | ρqᵗ | Total moisture density (conservative water variable) |
| $q^{t}$ | qᵗ | Total moisture mass fraction (vapor + liquid + ice) |
| $q^{v}$ | qᵛ | Vapor specific humidity |
| $q^{\ell}$ | qˡ | Liquid specific humidity |
| $q^{i}$ | qi | Ice specific humidity |
| $q^{v+}$ | qᵛ⁺ | Saturation specific humidity over liquid/ice (context-dependent) |
| $g$ | g | Gravitational acceleration |
| $\mathcal{R}$ | ℛ | Universal (molar) gas constant |
| $R^{d}$ | Rᵈ | Dry air gas constant |
| $R^{v}$ | Rᵛ | Water vapor gas constant |
| $R^{m}$ | Rᵐ | Mixture gas constant, function of $q$ |
| $c^{pd}$ | cᵖᵈ | Heat capacity of dry air at constant pressure |
| $c^{pv}$ | cᵖᵛ | Heat capacity of vapor at constant pressure |
| $c^l$ | cˡ | Heat capacity of the liquid phase (incompressible) |
| $c^i$ | cⁱ | Heat capacity of the ice phase (incompressible) |
| $c^{pm}$ | cᵖᵐ | Mixture heat capacity at constant pressure |
| $\theta$ | θ | Potential temperature |
| $\theta_0$ | θ₀ | Reference potential temperature (constant) |
| $\Pi$ | Π | Exner function, $\Pi = (p_r/p_0)^{R^{m}/c^{pm}}$ |
| $p$ | p | Pressure |
| $p_0$ | p₀ | Base (surface) reference pressure |
| $\rho_r(z)$ | ρᵣ | Density of a dry reference state at height $z$ |
| $\alpha_{r}$ | αᵣ | Specific volume of a dry reference state, $α_{r} = R^d θ_0 / p_r$ |
| $p_r(z)$ | pᵣ | Hydrostatic reference pressure at height $z$ |
| $\mathcal{L}^{l}$ | ℒˡ | Latent heat of vaporization |
| $b$ | b | Buoyancy |
| $p_h'$ | pₕ′ | Hydrostatic pressure anomaly, $∂_z p_h' = - ρ_r b$ |
| $p_n$ | pₙ | Nonhydrostatic pressure (projection/correction potential) |
| $\Delta t$ | Δt | Time step |
| $\Delta z$ | Δz | Vertical grid spacing |
Notes:
- Reference-state quantities use a subscript $r$ (e.g., $p_r$, $\rho_r$), following the Thermodynamics docs and code.
- Phase or mixture identifiers ($d$, $v$, $m$) appear as superscripts (e.g., $R^{d}$, $c^{pm}$), matching usage in the codebase (e.g.,
Rᵈ,cᵖᵐ). - Conservative variables are stored in ρᵣ-weighted form in the code (e.g.,
ρu,ρv,ρw,ρe,ρqᵗ). - Mapping to AtmosphereModel fields:
ρecorresponds tomodel.energy_density,ρqᵗtomodel.moisture_density, andqᵗtomodel.moisture_mass_fraction.