Notation
This page summarizes the mathematical and code notation used in NumericalEarth.jl, following the conventions established in Breeze.jl.
How the notation works
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),ˡᵈ(land) - Interface pair:
ᵃᵒ(atm–ocean),ᵃⁱ(atm–ice),ⁱᵒ(ice–ocean) - Direction:
ˣ/ʸ(spatial),ˢʷ/ˡʷ(shortwave / longwave) - Process:
ⁱⁿᵗ(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, ₚ penetrating) 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>.
Base flux symbols
| 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) |
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.
Superscript and subscript labels
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.
Superscript labels
| Label | Code | Meaning | Example |
|---|---|---|---|
| $v$ | ᵛ | water vapor | $\mathcal{Q}^v$ (latent heat flux) |
| $T$ | ᵀ | temperature / sensible | $\mathcal{Q}^T$ (sensible heat flux) |
| $c$ | ᶜ | condensate | $J^c$ (precipitation mass flux) |
| $S$ | 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) |
| $\mathrm{int}$ | ⁱⁿᵗ | interface | $T^{\mathrm{int}}$ (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) |
| $\mathrm{ld}$ | ˡᵈ | land | |
| $\mathrm{ao}$ | 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) |
Modifier arrows
| Symbol | Code | Tab completion | Meaning |
|---|---|---|---|
| ꜜ | ꜜ | \^downarrow | downwelling |
| ꜛ | ꜛ | \^uparrow | upwelling |
Subscript labels
| Label | Code | Meaning | Example |
|---|---|---|---|
| $t$ | ₜ | transmitted | $\mathscr{I}_{t}^{\mathrm{sw}}$ (transmitted shortwave) |
| $a$ | ₐ | absorbed | $\mathscr{I}_{a}^{\mathrm{lw}}$ (absorbed longwave) |
| $p$ | ₚ | penetrating | $\mathscr{I}_{p}^{\mathrm{sw}}$ (penetrating shortwave) |
| $\star$ | ★ | similarity theory scale | $u_\star$ (friction velocity) |
Atmosphere state variables
| 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) |
| $J^c$ | Jᶜ | condensate flux | Precipitation (condensate) mass flux (kg m⁻² s⁻¹) |
| $h_{b\ell}$ | h_bℓ | boundary layer height | Atmospheric boundary layer height (m) |
Ocean state variables
| 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) |
Sea ice state variables
| 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) |
Radiation properties
| Math | Code | Property | Description |
|---|---|---|---|
| $\sigma$ | σ | Stefan–Boltzmann constant | (W m⁻² K⁻⁴) |
| $\alpha$ | α | albedo | Surface reflectivity (–) |
| $\epsilon$ | ϵ | emissivity | Surface emissivity (–) |
Similarity theory / surface layer
| 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 (–) |
| $\zeta$ | ζ | stability parameter | $z / L_\star$ (–) |
| $\ell$ | ℓ | roughness length | Aerodynamic roughness length (m) |
| $\varkappa$ | ϰ | von Kármán constant | $\approx 0.4$ (–) |
Radiative fluxes
| 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) | | $\mathscr{I}_{p}^{\mathrm{sw}}$ | ℐₚˢʷ | penetrating shortwave | Shortwave penetrating into the ocean interior (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, ₚ) with a superscript band.
Turbulent interface fluxes
| 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) |
Net ocean fluxes
| 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) |
Thermodynamic properties
| 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) |
CF standard name mapping
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 |
Typing Unicode symbols in Julia
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) |
★ | \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) |