Notation and Symbols
The governing equations and the PROPHET equations use the notation of the papers they follow ([1] for the dynamical core). The source code uses ASCII- and Unicode-flavored names that encode where a field lives and which subdomain it belongs to. This page is the bridge between the two.
Staggered-grid prefixes
The vertical grid is staggered: some quantities live at cell centers, others at cell faces (the interfaces between centers). The leading character of a field or operator name says which.
| Prefix | Meaning | Example |
|---|---|---|
ᶜ | Cell center | ᶜρ, ᶜp, ᶜK, ᶜinterp, ᶜdivᵥ |
ᶠ | Cell face (vertical interface) | ᶠu₃, ᶠu³, ᶠinterp, ᶠgradᵥ |
The state vector mirrors this split: Y.c holds center variables and Y.f holds face variables, and the same convention applies to the cache p.
For operators the prefix names the output location, so ᶜinterp interpolates faces to centers and ᶠinterp interpolates centers to faces. A ᵥ subscript marks a vertical (finite-difference) operator and an ₕ subscript a horizontal (spectral-element) one: ᶜdivᵥ versus divₕ. A leading w on a horizontal operator marks the weak form, as in wdivₕ and wcurlₕ.
Subdomain superscripts
PROPHET partitions each grid cell into subdomains: the grid mean, one or more drafts (updrafts), and the environment.
| Superscript | Meaning | Example |
|---|---|---|
| (none) | Grid mean | ᶜρ, ᶜu |
ʲ | A single draft (subdomain j) | ᶜρaʲ, ᶠu₃ʲ, ᶜmseʲ |
ʲs | The tuple over all drafts | Y.c.sgsʲs, ᶜρaʲs, ᶜuʲs |
⁰ | The environment | ᶜmse⁰, ᶜq_tot⁰, ᶠu₃⁰ |
The environment is not stored: it is recovered as the residual of the grid mean minus the drafts.
Vector components
Vectors are stored in the covariant or contravariant bases of the reference element rather than as physical components. Subscripts denote covariant components and superscripts contravariant ones.
| Symbol | Meaning |
|---|---|
uₕ | Horizontal covariant velocity (Covariant12Vector) |
u₃ | Vertical covariant velocity (Covariant3Vector) |
u³ | Third contravariant velocity component |
C1, C2, C12, C3, C123 | Covariant vector constructors |
CT1, CT2, CT12, CT3, CT123 | Contravariant vector constructors |
UVec, VVec, WVec, UV, UVW | Physical (local orthonormal) vectors |
Contravariant velocity components have units of [1/s], not [m/s]: they are velocities per unit reference-element length.
Prognostic variables
| Paper symbol | Code name | Location | Description |
|---|---|---|---|
| $\rho$ | ρ | center | Moist air density [kg/m³]. |
| $\boldsymbol{u}_h$ | uₕ | center | Horizontal covariant velocity. |
| $u_3$ | u₃ | face | Vertical covariant velocity. |
| $\rho e_{tot}$ | ρe_tot | center | Total energy density [J/m³]. |
| $\rho q_t$ | ρq_tot | center | Total water content [kg/m³]. |
| $\rho q_l^{cl}$ | ρq_lcl | center | Cloud liquid water content [kg/m³]. |
| $\rho q_i^{cl}$ | ρq_icl | center | Cloud ice content [kg/m³]. |
| $\rho q_r$ | ρq_rai | center | Rain water content [kg/m³]. |
| $\rho q_s$ | ρq_sno | center | Snow content [kg/m³]. |
| $\rho n_l$ | ρn_lcl | center | Cloud droplet number concentration [1/m³]. |
| $\rho n_r$ | ρn_rai | center | Rain drop number concentration [1/m³]. |
| $\rho e_{tke}$ | ρtke | center | Turbulence kinetic energy density [J/m³]. |
Which of these exist depends on the configuration: the moisture and precipitation variables are added by the microphysics model, and ρtke by the turbulence-convection model. Note the code writes cloud condensate as lcl/ icl (liquid-cloud, ice-cloud) to distinguish it from precipitating species.
Draft variables live in Y.c.sgsʲs.:(j) and are specific (not density-weighted), except for the effective density itself:
| Paper symbol | Code name | Description |
|---|---|---|
| $\hat{\rho}^j$ | ρa | Effective density $\rho^j a^j$ [kg/m³]. |
| $h_s^j$ | mse | Specific moist static energy [J/kg]. |
| $q_t^j$ | q_tot | Specific total water [kg/kg]. |
| $u_3^j$ | u₃ | Vertical covariant velocity of the draft (a face variable). |
With a slab-ocean surface, the prognostic surface state is Y.sfc.
Derived and cached quantities
| Paper symbol | Code name | Description |
|---|---|---|
| $p$ | ᶜp | Air pressure [Pa]. |
| $K$ | ᶜK | Specific kinetic energy [J/kg]. |
| $\Phi$ | ᶜΦ | Geopotential [m²/s²]. |
| $\Pi$ | ᶜΠ | Exner function [-]. |
| $\tilde{\boldsymbol u}$ | ᶠu³ | Mass-weighted face velocity (contravariant). |
| $\bar{\boldsymbol u}$ | ᶜu | Cell-center reconstruction of the velocity. |
| $\rho^j$ | ᶜρʲs | Draft air densities in kg/m³; the area fraction is a^j = ρa/ρʲ. |
| $T$ | ᶜT | Air temperature [K]. |
| $h_{tot}$ | ᶜh_tot | Total specific enthalpy [J/kg]. |
| $K_h$, $K_u$ | ᶠK_h, ᶠK_u | Eddy diffusivity and viscosity [m²/s]. |
| $\ell$ | ᶜl_mix | Mixing length [m]. |
Tendencies carry a ₜ subscript: Yₜ is the tendency of the state, and Yₜ.c.ρe_tot is the tendency of total energy.
Common suffixes and shorthands
| Suffix / name | Meaning |
|---|---|
_nonneg | A copy clipped at zero, e.g. ᶜq_tot_nonneg. |
ʲs / ⁰ | Cache fields carry the same subdomain suffixes as the state, e.g. ᶜTʲs, ᶜT⁰, ᶜq_liq⁰. |
ᶜspecific | Conversion from a density-weighted variable to a specific one. |
FT | The working float type (Float32 or Float64). |
Y, Yₜ, p, t | State, tendency, cache, and time; see the Glossary. |
The operator shorthands (ᶜinterp, ᶠwinterp, ᶜadvdivᵥ, ᶠupwind1, …) are defined and documented in src/utils/abbreviations.jl; the governing equations page maps the mathematical operators onto their ClimaCore implementations.