This page summarizes the mathematical and code notation used in NumericalEarth.jl, following the conventions established in Breeze.jl.
Variable names are built by combining a base symbol with superscripts and, occasionally, a short plain-text tag.
Base symbols are single characters (often script letters) that identify the physical category of a quantity — for example, 𝒬 for heat flux, ℐ for radiative intensity, J for mass flux, and τ for kinematic momentum flux.
Superscripts refine the meaning in several ways:
- Phase or species:
ᵛ (vapor), ˡ (liquid), ⁱ (ice), ᶜ (condensate) - Component:
ᵃᵗ (atmosphere), ᵒᶜ (ocean), ˢⁱ (sea ice), ˡᵃ (land) - Interface pair:
ᵃᵒ (atm–ocean), ᵃⁱ (atm–ice), ⁱᵒ (ice–ocean) - Direction:
ˣ / ʸ (spatial), ˢʷ / ˡʷ (shortwave / longwave) - Process:
ⁱⁿ (interface), ᶠʳᶻ (frazil)
Modifier arrows ꜜ (\^downarrow) and ꜛ (\^uparrow) denote downwelling and upwelling directions in radiative fluxes.
Subscripts encode radiative process (ₜ transmitted, ₐ absorbed) and the similarity-theory scale ★.
For example, 𝒬ᵛ is the latent (vapor) heat flux, ℐꜜˢʷ is the downwelling shortwave radiative intensity, and τˣ is the zonal kinematic momentum flux.
In Julia code, superscripts are entered with Unicode (e.g. \scrQ<tab> → 𝒬, then \^v<tab> → ᵛ). The modifier arrows ꜜ and ꜛ are entered with \^downarrow<tab> and \^uparrow<tab>.
| Math | Code | Tab completion | Meaning |
|---|
| $\mathcal{Q}$ | 𝒬 | \scrQ | Heat flux (W m⁻²) |
| $\mathscr{I}$ | ℐ | \scrI | Radiative intensity (W m⁻²) |
| $J$ | J | | Mass flux (kg m⁻² s⁻¹) |
| $\tau$ | τ | \tau | Kinematic momentum flux (m² s⁻²) |
| $\mathcal{L}$ | ℒ | \scrL | Latent heat (J kg⁻¹) |
| $M$ | M | | Layer-integrated mass per area (kg m⁻²) |
Note: $\tau^x$ (τˣ) is the kinematic momentum flux (stress divided by density). The mass-weighted stress is $\rho \tau^x$ (ρτˣ, in N m⁻²).
These base symbols are combined with superscript and subscript labels (documented below) to form specific variable names.
Superscripts and subscripts are used systematically to label physical quantities. Superscripts generally denote the type or phase of a quantity, while subscripts denote the component or location.
| Label | Code | Meaning | Example |
|---|
| $v$ | ᵛ | water vapor | $\mathcal{Q}^v$ (latent heat flux) |
| $T$ | ᵀ | temperature / sensible | $\mathcal{Q}^T$ (sensible heat flux) |
| $\mathrm{rn}$ | ʳⁿ | rain | $J^{\mathrm{rn}}$ (rainfall) |
| $\mathrm{sn}$ | ˢⁿ | snow | $J^{\mathrm{sn}}$ (snowfall) |
| $S$ | ˢ | salinity | $J^S$ (salinity flux) |
| $i$ | ⁱ | ice | $\mathcal{L}^i$ (latent heat of sublimation) |
| $\ell$ | ˡ | liquid | $\mathcal{L}^\ell$ (latent heat of vaporization) |
| $p$ | ᵖ | constant pressure | $c^{pm}$ (moist isobaric heat capacity) |
| $m$ | ᵐ | mixture (moist air) | $c^{pm}$ (moist isobaric heat capacity) |
| $d$ | ᵈ | dry (air) | $c^{pd}$ (dry air heat capacity) |
| $D$ | ᴰ | drag | $C^D$ (drag coefficient) |
| $\mathrm{in}$ | ⁱⁿ | interface | $T^{\mathrm{in}}$ (interface temperature) |
| $\mathrm{frz}$ | ᶠʳᶻ | frazil | $\mathcal{Q}^{\mathrm{frz}}$ (frazil heat flux) |
| $x$ | ˣ | zonal / x-direction | $\tau^x$ (zonal kinematic stress) |
| $y$ | ʸ | meridional / y-direction | $\tau^y$ (meridional kinematic stress) |
| $\mathrm{at}$ | ᵃᵗ | atmosphere | $\rho^{\mathrm{at}}$ (air density) |
| $\mathrm{oc}$ | ᵒᶜ | ocean | $\rho^{\mathrm{oc}}$ (ocean reference density) |
| $\mathrm{si}$ | ˢⁱ | sea ice | $h^{\mathrm{si}}$ (sea ice thickness) |
| $\mathrm{la}$ | ˡᵃ | land | $M^{\mathrm{la}}$ (land water mass per area) |
| $\mathrm{ao}$ | ᵃᵒ | atmosphere–ocean interface | $\mathcal{Q}^{\mathrm{ao}}$ (atm–ocean heat flux) |
| $\mathrm{ai}$ | ᵃⁱ | atmosphere–ice interface | $\mathcal{Q}^{\mathrm{ai}}$ (atm–ice heat flux) |
| $\mathrm{io}$ | ⁱᵒ | ice–ocean interface | $\mathcal{Q}^{\mathrm{io}}$ (ice–ocean heat flux) |
| $\mathrm{sw}$ | ˢʷ | shortwave | $\mathscr{I}$ ꜜ ${}^{\mathrm{sw}}$ (downwelling shortwave) |
| $\mathrm{lw}$ | ˡʷ | longwave | $\mathscr{I}$ ꜜ ${}^{\mathrm{lw}}$ (downwelling longwave) |
| Symbol | Code | Tab completion | Meaning |
|---|
| ꜜ | ꜜ | \^downarrow | downwelling |
| ꜛ | ꜛ | \^uparrow | upwelling |
| Label | Code | Meaning | Example |
|---|
| $t$ | ₜ | transmitted | $\mathscr{I}_{t}^{\mathrm{sw}}$ (transmitted shortwave) |
| $a$ | ₐ | absorbed | $\mathscr{I}_{a}^{\mathrm{lw}}$ (absorbed longwave) |
| $\star$ | ★ | similarity theory scale | $u_\star$ (friction velocity) |
| Math | Code | Property | Description |
|---|
| $T$ | T | temperature | Air temperature (K) |
| $p$ | p | pressure | Air pressure (Pa) |
| $q$ | q | specific humidity | Mass mixing ratio of water vapor (kg kg⁻¹) |
| $u$ | u | zonal velocity | Eastward wind component (m s⁻¹) |
| $v$ | v | meridional velocity | Northward wind component (m s⁻¹) |
| $\mathscr{I}_\downarrow^{\mathrm{sw}}$ | ℐꜜˢʷ | downwelling shortwave | Downwelling shortwave radiation (W m⁻²) |
| $\mathscr{I}_\downarrow^{\mathrm{lw}}$ | ℐꜜˡʷ | downwelling longwave | Downwelling longwave radiation (W m⁻²) |
| $h_{b\ell}$ | h_bℓ | boundary layer height | Atmospheric boundary layer height (m) |
| $pᵛ⁺$ | $pᵛ⁺$ | saturation vapor pressure | Vapor pressure at saturation (Pa) |
| $qᵛ⁺$ | qᵛ⁺ | saturation specific humidity | Specific humidity at saturation, $q^{v+}(T)$ (kg kg⁻¹) |
| $qˢ$ | q | surface specific humidity | Specific humidity at the interface; set by the humidity model (β·qᵛ⁺ for FractionalHumidity, a vapor-flux balance for SkinHumidity) (kg kg⁻¹) |
| Math | Code | Property | Description |
|---|
| $T$ | temperature | bulk land temperature | Prognostic land-column temperature (K) |
| $M^{\mathrm{la}}$ | water_storage | land water | Prognostic land water mass per area (kg m⁻²) |
| $M^{\mathrm{la}\!+}$ | maximum_water_storage | maximum land water | Bucket capacity; soil-science "field capacity" (kg m⁻²) |
| $𝒮$ | saturation | surface saturation | Continuous land surface saturation $\mathrm{clamp}(Mˡᵃ/Mˡᵃ⁺, 0, 1)$; the interface humidity models derive their availability $β$ from it (–) |
| $𝒮ᶜ$ | critical_saturation | critical saturation | Saturation above which the surface evaporates at full efficiency, for CriticalSaturation (–) |
| $T^{\mathrm{deep}}$ | deep_temperature | deep climatological temperature | Prescribed deep/climatological target temperature for force-restore (K) |
| $τ^{\mathrm{deep}}$ | deep_time_scale | deep-restore time scale | Time scale of surface relaxation toward $T^{\mathrm{deep}}$ (s) |
| $d$ | surface_thickness | surface thickness | Thickness of the dry surface layer through which soil vapor diffuses, for SkinHumidity (m) |
| $κ^q$ | vapor_diffusivity | soil vapor diffusivity | Vapor mass diffusivity in the surface soil layer, for SkinHumidity (kg m⁻¹ s⁻¹) |
| Math | Code | Property | Description |
|---|
| $T$ | T | temperature | Ocean potential temperature (ᵒC or K) |
| $S$ | S | salinity | Practical salinity (g kg⁻¹) |
| $u$ | u | zonal velocity | Eastward ocean velocity (m s⁻¹) |
| $v$ | v | meridional velocity | Northward ocean velocity (m s⁻¹) |
| $\rho^{\mathrm{oc}}$ | ρᵒᶜ | reference density | Ocean reference density (kg m⁻³) |
| $c^{\mathrm{oc}}$ | cᵒᶜ | heat capacity | Ocean heat capacity (J kg⁻¹ K⁻¹) |
| Math | Code | Property | Description |
|---|
| $h^{\mathrm{si}}$ | hˢⁱ | ice thickness | Sea ice thickness (m) |
| $\aleph$ | ℵ | ice concentration | Areal fraction of ice cover (–) |
| $S^{\mathrm{si}}$ | Sˢⁱ | ice salinity | Sea ice bulk salinity (g kg⁻¹) |
| Math | Code | Property | Description |
|---|
| $\sigma$ | σ | Stefan–Boltzmann constant | (W m⁻² K⁻⁴) |
| $\alpha$ | α | albedo | Surface reflectivity (–) |
| $\epsilon$ | ϵ | emissivity | Surface emissivity (–) |
| Math | Code | Property | Description |
|---|
| $u_\star$ | u★ | friction velocity | Surface friction velocity (m s⁻¹) |
| $\theta_\star$ | θ★ | temperature scale | Flux characteristic temperature (K) |
| $q_\star$ | q★ | humidity scale | Flux characteristic specific humidity (kg kg⁻¹) |
| $b_\star$ | b★ | buoyancy scale | Flux characteristic buoyancy (m s⁻²) |
| $L_\star$ | L★ | Obukhov length | Monin–Obukhov length scale (m) |
| $C^D$ | Cᴰ | drag coefficient | Bulk transfer coefficient for momentum (–) |
| $\psi$ | ψ | stability function | Integrated stability correction (–) |
| $\Psi$ | Ψ | interface state | Aggregate interface state (an AbstractInterfaceState) carried through the similarity-theory fixed-point solver compute_interface_state |
| $\zeta$ | ζ | stability parameter | $z / L_\star$ (–) |
| $\ell$ | ℓ | roughness length | Aerodynamic roughness length (m) |
| $\ell^\mathrm{m}$ | ℓᵐ | momentum roughness length | Aerodynamic momentum roughness length (m) |
| $\ell^\mathrm{s}$ | ℓˢ | scalar roughness length | Aerodynamic scalar roughness length (m) |
| $\varkappa$ | ϰ | von Kármán constant | $\approx 0.4$ (–) |
Note the case distinction: lowercase $\psi$ (ψ) is the stability function, while capital $\Psi$ (Ψ) is the aggregate interface-state object.
| Math | Code | Property | Description |
|---|
| $\mathscr{I}_\downarrow^{\mathrm{sw}}$ | ℐꜜˢʷ | downwelling shortwave | Downwelling shortwave radiation (W m⁻²) |
| $\mathscr{I}_\downarrow^{\mathrm{lw}}$ | ℐꜜˡʷ | downwelling longwave | Downwelling longwave radiation (W m⁻²) |
| $\mathscr{I}_\uparrow^{\mathrm{lw}}$ | ℐꜛˡʷ | upwelling longwave | Emitted longwave radiation (W m⁻²) |
| $\mathscr{I}_{t}^{\mathrm{sw}}$ | ℐₜˢʷ | transmitted shortwave | Shortwave passing through the surface, $(1-\alpha) \mathscr{I}_\downarrow^{\mathrm{sw}}$ (W m⁻²) | | $\mathscr{I}_{a}^{\mathrm{lw}}$ | ℐₐˡʷ | absorbed longwave | Longwave absorbed at the surface, $\epsilon \mathscr{I}_\downarrow^{\mathrm{lw}}$ (W m⁻²) |
Radiative fluxes use $\mathscr{I}$ (ℐ, for "intensity") with a modifier arrow (ꜜ/ꜛ for downwelling/upwelling) and superscript band (ˢʷ/ˡʷ). Derived radiative quantities use a subscript process label (ₜ, ₐ) with a superscript band.
| Math | Code | Property | Description |
|---|
| $\mathcal{Q}^v$ | 𝒬ᵛ | latent heat flux | Turbulent latent heat flux (W m⁻²) |
| $\mathcal{Q}^T$ | 𝒬ᵀ | sensible heat flux | Turbulent sensible heat flux (W m⁻²) |
| $J^v$ | Jᵛ | water vapor flux | Turbulent mass flux of water vapor (kg m⁻² s⁻¹) |
| $\tau^x$ | τˣ | zonal kinematic stress | Kinematic zonal momentum flux (m² s⁻²) |
| $\tau^y$ | τʸ | meridional kinematic stress | Kinematic meridional momentum flux (m² s⁻²) |
| $\rho \tau^x$ | ρτˣ | zonal wind stress | Mass-weighted zonal stress (N m⁻²) |
| $\rho \tau^y$ | ρτʸ | meridional wind stress | Mass-weighted meridional stress (N m⁻²) |
| Math | Code | Property | Description |
|---|
| $J^T$ | Jᵀ | temperature flux | Net ocean temperature flux (K m s⁻¹) |
| $J^S$ | Jˢ | salinity flux | Net ocean salinity flux (g kg⁻¹ m s⁻¹) |
| $\mathcal{Q}^{\mathrm{frz}}$ | 𝒬ᶠʳᶻ | frazil heat flux | Heat released by frazil ice formation (W m⁻²) |
| Math | Code | Property | Description |
|---|
| $J^{\mathrm{rn}}$ | Jʳⁿ | rain freshwater flux | Rain mass flux at the surface (kg m⁻² s⁻¹) |
| $J^{\mathrm{sn}}$ | Jˢⁿ | snow freshwater flux | Snow mass flux at the surface (kg m⁻² s⁻¹) |
| Math | Code | Property | Description |
|---|
| $\mathcal{L}^\ell$ | ℒˡ | latent heat of vaporization | Liquid-phase latent heat (J kg⁻¹) |
| $\mathcal{L}^i$ | ℒⁱ | latent heat of sublimation | Ice-phase latent heat (J kg⁻¹) |
| $c^{pm}$ | cᵖᵐ | moist air heat capacity | Moist isobaric specific heat (J kg⁻¹ K⁻¹) |
| $c^{pd}$ | cᵖᵈ | dry air heat capacity | Dry-air isobaric specific heat (J kg⁻¹ K⁻¹) |
| $\rho^{\mathrm{at}}$ | ρᵃᵗ | air density | Atmospheric air density (kg m⁻³) |
| $\varepsilon^{\mathrm{dv}}$ | $εᵈᵛ$ | vapor / dry-air gas-constant ratio | $εᵈᵛ = R_v / R_d$ (so $(εᵈᵛ)^{-1} = R_d / R_v ≈ 0.622$ is the conventional ε in $q = ε e / p$) (–) |
The following table maps code variable names to their CF standard names where applicable.
| Code | CF standard name |
|---|
T (atm) | air_temperature |
T (ocn) | sea_water_potential_temperature |
S | sea_water_practical_salinity |
u (atm) | eastward_wind |
v (atm) | northward_wind |
q | specific_humidity |
p | air_pressure |
ℐꜜˢʷ | surface_downwelling_shortwave_flux_in_air |
ℐꜜˡʷ | surface_downwelling_longwave_flux_in_air |
𝒬ᵛ | surface_upward_latent_heat_flux |
𝒬ᵀ | surface_upward_sensible_heat_flux |
Jᵛ | water_evapotranspiration_flux |
ρτˣ | surface_downward_eastward_stress |
ρτʸ | surface_downward_northward_stress |
hˢⁱ | sea_ice_thickness |
ℵ | sea_ice_area_fraction |
Most symbols can be entered in the Julia REPL and in editors with Julia support by typing a LaTeX-like abbreviation followed by <tab>. The table below collects the less obvious completions used in this notation.
| Symbol | Tab completion | Description |
|---|
𝒬 | \scrQ | Script Q (heat flux) |
ℐ | \scrI | Script I (radiative intensity) |
ℒ | \scrL | Script L (latent heat) |
τ | \tau | Tau (kinematic stress) |
ρ | \rho | Rho (density) |
σ | \sigma | Sigma (Stefan–Boltzmann constant) |
α | \alpha | Alpha (albedo) |
ϵ | \epsilon | Epsilon (emissivity) |
ℵ | \aleph | Aleph (ice concentration) |
ϰ | \varkappa | Varkappa (von Kármán constant) |
Ψ | \Psi | Capital Psi (interface state) |
★ | \bigstar | Star (similarity-theory scale) |
ꜜ | \^downarrow | Modifier down arrow (downwelling) |
ꜛ | \^uparrow | Modifier up arrow (upwelling) |
ᵛ | \^v | Superscript v |
ᵀ | \^T | Superscript T |
ˢ | \^s | Superscript s |
ʷ | \^w | Superscript w |
ⁱ | \^i | Superscript i |
ˡ | \^l | Superscript l |
ᵖ | \^p | Superscript p |
ᵐ | \^m | Superscript m |
ᵈ | \^d | Superscript d |
ᴰ | \^D | Superscript D |
ˣ | \^x | Superscript x |
ʸ | \^y | Superscript y |
ᵃ | \^a | Superscript a |
ᵗ | \^t | Superscript t |
ᵒ | \^o | Superscript o |
ᶜ | \^c | Superscript c |
ⁿ | \^n | Superscript n |
ᶠ | \^f | Superscript f |
ʳ | \^r | Superscript r |
ᶻ | \^z | Superscript z |
ₜ | \_t | Subscript t (transmitted) |
ₐ | \_a | Subscript a (absorbed) |
ₚ | \_p | Subscript p (penetrating) |