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),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)
Component superscripts are used only in cross-component context — wherever a variable appears alongside variables from other components, as in interface computations or coupled-model discussions (Tˡᵃ next to Tᵃᵗ and Tⁱⁿ). Within a single component's own namespace the bare symbol is used: the land model's prognostic state is (; T, M) (as returned by prognostic_fields(land)), just as the ocean's is (; u, v, w, T, S).
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>.
Base 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) |
| $M$ | 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.
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) |
| $\mathrm{rn}$ | ʳⁿ | rain | $J^{\mathrm{rn}}$ (rainfall) |
| $\mathrm{sn}$ | ˢⁿ | snow | $J^{\mathrm{sn}}$ (snowfall) |
| $S$ | ˢ | salinity | $J^S$ (salinity flux) |
| $w$ | ʷ | freshwater | $J^w$ (freshwater volume flux per unit area) |
| $h$ | ʰ | height (roughness element) | $\sigma^h$ (height standard deviation) |
| $i$ | 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) |
| $\mathrm{ae}$ | ᵃᵉ | air entry | $\alpha^{\mathrm{ae}}$ (inverse air-entry head) |
| $\mathrm{oc}$ | 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) |
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) |
| $\star$ | ★ | similarity theory scale | $u_\star$ (friction velocity) |
| $b$ | b | boundary value, or background | $T_b$ (interface temperature), $u_b$ (background bottom 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) |
| $h_{b\ell}$ | 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) |
Land state variables and parameters
Bare symbols below are the land model's internal names; in cross-component context they take the ˡᵃ superscript (Tˡᵃ, Mˡᵃ, Mˡᵃ⁺) per the component-superscript rule above.
| Math | Code | Property | Description |
|---|---|---|---|
| $T$ | temperature | ground temperature | Prognostic land-column temperature (K) |
| $M$ | water_storage | land water | Prognostic land water mass per area (kg m⁻²) |
| $M^{+}$ | maximum_water_storage | maximum land water | Bucket capacity; soil-science "field capacity" (kg m⁻²) |
| $𝒮$ | saturation | surface saturation | Continuous land surface saturation $\mathrm{clamp}(M/M⁺, 0, 1)$; 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 (–) |
| $𝒮ᶜ$ | dry_layer_onset_saturation | dry-layer onset saturation | Saturation below which a dry surface layer forms, for StorageBasedDryLayerDepth; shares the symbol $𝒮ᶜ$ with critical_saturation above (–) |
| $T^{\mathrm{deep}}$ | 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) |
| $c^{\mathrm{dry}}$ | dry_heat_capacity | dry areal heat capacity | Areal heat capacity of the water-free slab; a Number or an AbstractField (J m⁻² K⁻¹) |
| $c^l$ | liquid_heat_capacity | liquid heat capacity | Specific heat capacity of slab liquid water (J kg⁻¹ K⁻¹) |
| $c^{\mathrm{la}}$ | – | land areal heat capacity | Water-dependent areal heat capacity $c^{\mathrm{la}} = c^{\mathrm{dry}} + c^l M$ (J m⁻² K⁻¹) |
| $d$ | 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) |
| $\chi^{\mathrm{sand}}$ | sand | soil sand fraction | Mass fraction of sand grains in the mineral (non-organic) solid matrix (kg kg⁻¹) |
| $\chi^{\mathrm{silt}}$ | silt | soil silt fraction | Mass fraction of silt grains in the mineral (non-organic) solid matrix (kg kg⁻¹) |
| $\chi^{\mathrm{clay}}$ | clay | soil clay fraction | Mass fraction of clay grains in the mineral (non-organic) solid matrix (kg kg⁻¹) |
| $\chi^{\mathrm{soc}}$ | SOC | soil organic carbon concentration | Mass fraction of organic carbon in the solid matrix (kg kg⁻¹) |
| $\rho^{\mathrm{soil}}$ | ρ_soil | soil bulk dry density | Bulk dry density of the soil within each vertical layer (kg m⁻³) |
| $\rho^{\mathrm{soc}}$ | ρ_soc | soil organic carbon density | Bulk density of organic material within each vertical layer (kg m⁻³) |
Variably-saturated slab land
Symbols introduced by VariablySaturatedHydrology, WaterCoupledEnergy, and DryLayerHumidity.
| Math | Code | Property | Description |
|---|---|---|---|
| $h^{\mathrm{la}}$ | slab_depth | depth of prognostic land | Vertical thickness of the integrated land slab, from $z_b$ to $z_s$ (m) |
| $\nu$ | porosity | soil porosity | Total pore fraction (–) |
| $\theta^l$ | – | pore liquid fraction | Physical liquid-filled pore fraction; surface physics consumes this (–) |
| $\vartheta^l$ | – | augmented liquid fraction | Conservative storage variable $= \theta^l + \max(\Pi, 0)/h^{\mathrm{ss}}$; allows $M > M⁺$ saturated overflow (–) |
| $\theta^r$ | residual_liquid_fraction | residual liquid fraction | Minimum liquid-filled pore fraction (–) |
| $𝒮$ | saturation | effective saturation | Effective (relative) saturation $𝒮 = \mathrm{clamp}\!\left((\theta^l - \theta^r)/(\nu - \theta^r),\, 0,\, 1\right)$; the humidity availability and the front depth $\delta^v$ derive from it (–) |
| $h^{\mathrm{ss}}$ | storage_height | storage height | Saturated storage height — the head built per unit fractional over-saturation; reciprocal of the specific storage ($1/S_s$) (m) |
| $\Pi$ | – | soil pressure head | Matric/pressure head; $\Pi \le 0$ unsaturated, $\Pi > 0$ saturated overflow (m) |
| $\Pi^d$ | deep_pressure_head | deep pressure head | Pressure head of the deep reservoir below the slab, passed to the deep-flux closure (m) |
| $h$ | – | hydraulic head | $h = z + \Pi$ (m) |
| $K$ | – | hydraulic conductivity | Darcy conductivity (m s⁻¹) |
| $K_0$ | matching_point_conductivity | matching-point conductivity | Conductivity the Mualem–van Genuchten curve reaches at $𝒮 = 1$; a matrix value when it comes from a pedotransfer function, excluding macropore flow (m s⁻¹) |
| $K^{+}$ | saturated_conductivity | saturated hydraulic conductivity | Macropore-inclusive Darcy conductivity of saturated soil, for an infiltration cap (m s⁻¹) |
| $\alpha^{\mathrm{ae}}$ | inverse_air_entry_head | inverse air-entry head | Reciprocal of the air-entry pressure head, the suction at which the largest pores begin to empty (m⁻¹) |
| $\mathscr{n}$ | pore_size_uniformity | pore-size uniformity | van Genuchten retention exponent; large $\mathscr{n}$ drains over a narrow band of suctions, $\mathscr{n} \to 1$ over orders of magnitude. Sets $\mathscr{m} = 1 - 1/\mathscr{n}$ (–) |
| $\eta^K$ | ηᴷ, pore_connectivity_exponent | pore-connectivity exponent | Mualem exponent on saturation in $K(\mathcal S)$ (–) |
| $T_0$ | reference_temperature | viscosity reference temperature | Temperature at which $K_0$ holds and the viscosity factor $\Theta(T)$ on $K$ is unity (K) |
| $J^{Es}$ | surface_energy_flux | surface energy flux | Signed surface energy flux, positive upward (out of the slab) (W m⁻²) |
| $J^{lb}$ | deep_liquid_flux | deep-boundary liquid flux | Liquid mass flux across the slab bottom, positive upward (into the slab, capillary rise / groundwater return); drainage is $J^{lb} < 0$ (kg m⁻² s⁻¹) |
| $J^{ls}$ | surface_liquid_flux | surface liquid flux | Liquid mass flux at the surface $J^{ls} = -P^l + R^{\mathrm{sfc}}$, positive upward (out of the slab); infiltration is $J^{ls} < 0$ (kg m⁻² s⁻¹) |
| $R^{\mathrm{sfc}}$ | surface_runoff | surface runoff | Liquid input rejected at the surface, $\ge 0$ (kg m⁻² s⁻¹) |
| $R^{\mathrm{lat}}$ | subsurface_runoff | subsurface runoff | Lateral storage export, $\ge 0$ (kg m⁻² s⁻¹) |
| $\kappa^T$ | thermal_conductivity | thermal conductivity | Effective ground thermal conductivity (W m⁻¹ K⁻¹) |
| $\Lambda^{\mathrm{deep}}$ | deep_conductance | deep energy conductance | Force-restore deep energy conductance (W m⁻² K⁻¹); see also $τ^{\mathrm{deep}}$ |
| $T_r$ | reference_temperature | reference temperature | Reference temperature for internal energy $e^l(T) = c^l (T - T_r)$ (K) |
| $T^{\mathrm{in}}$ | – | interface temperature | Atmosphere-facing skin temperature, $T^{\mathrm{in}}$ (K) |
| $q^{\mathrm{in}}$ | – | interface specific humidity | Atmosphere-facing skin humidity, $q^{\mathrm{in}}$ (kg kg⁻¹) |
| $T^e$ | – | dry-layer temperature | Diagnostic temperature at the dry layer (K) |
| $q^e$ | – | dry-layer specific humidity | Vapor source humidity at the dry layer (kg kg⁻¹) |
| $\delta^v$ | dry_layer_depth | dry-layer depth | Dry-layer thickness through which vapor diffuses, diagnostic of $𝒮$ (m) |
| $\chi$ | – | blend coefficient | $\chi = \mathrm{clamp}(\delta^v/\ell^T, 0, 1)$; weights $T^e$ between $T^{\mathrm{in}}$ and $T^{\mathrm{la}}$ (–) |
| $\eta$ | dry_layer_exponent | front-depth exponent | Exponent in $\delta^v = \delta^v_{max}[1 - \min(𝒮/𝒮^c, 1)]^\eta$ (–) |
| $\ell^T$ | thermal_exchange_depth, exchange_depth | thermal exchange depth | Depth over which $\Lambda^{\mathrm{in}} = \kappa^T/\ell^T$ couples $T^{\mathrm{la}}$ to $T^{\mathrm{in}}$ (m) |
| $D^v$ | molecular_diffusivity | vapor diffusivity in air | Molecular vapor diffusivity in air (m² s⁻¹) |
| $w^d$ | – | dry-layer piston velocity | $w^d = D^v_{eff}/\max(\delta^v, \delta^v_{min})$ (m s⁻¹) |
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) |
| $u_b$ | ub | background bottom velocity | Unresolved velocity, mostly tidal, added in quadrature to the resolved speed in the bottom drag (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 (–) |
| $\Psi$ | Ψ | interface state | Aggregate interface state (an AbstractInterfaceState) carried through the similarity-theory fixed-point solver compute_interface_state |
| $\zeta$ | ζ | stability parameter | $z / L_\star$ (–) |
| $\Delta h^d$ | Δhᵈ | displaced profile height | Height $\Delta h - d$ above the zero-plane displacement at which the similarity profiles are evaluated (m) |
| $\ell$ | 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.
Surface morphometry
Roughness elements — buildings or plant canopy — that set the aerodynamic parameters $\ell^\mathrm{m}$ and $d$ of a land surface.
| Math | Code | Property | Description |
|---|---|---|---|
| $h$ | h | element height | Mean height of the roughness elements (m) |
| $h^\mathrm{max}$ | hᵐᵃˣ | maximum element height | Tallest roughness element in the cell (m) |
| $\sigma^h$ | σʰ | height standard deviation | Spread of the roughness-element heights (m) |
| $\lambda^p h$ | gross_building_height | gross building height | Mean element height over all cells, built or not, $\lambda^p h$ — the digital-surface lift (m) |
| $d$ | d | zero-plane displacement | Displacement height of the roughness sublayer (m) |
| $d^h$ | dʰ | displacement ratio | Packing displacement $d/h$ of the obstacle-array fit, before the height-spread correction (–) |
| $\lambda^p$ | λᵖ | plan-area index | Ground-plan area covered by the elements, per unit ground area (–) |
| $\lambda^f$ | λᶠ | frontal-area index | Windward face area of the elements, per unit ground area (–) |
| $\lambda^{p\,\mathrm{min}}$ | λᵖᵐⁱⁿ | plan-area index floor | Below this $\lambda^p$ a cell reduces to bare soil (–) |
| $\ell^\mathrm{soil}$ | ℓˢᵒⁱˡ | bare-soil roughness length | Momentum roughness length where the plan-area index vanishes (m) |
| $A$ | A | array constant | Packing constant of the Macdonald displacement fit (–) |
| $\beta$ | β | drag correction factor | Lumped drag correction of the Macdonald drag partition (–) |
| $X$ | X | displacement parameter | $(\sigma^h + h)/h^\mathrm{max} \in [0, 1]$, argument of the Kanda displacement fit (–) |
| $Y$ | Y | height-spread parameter | $\lambda^p \sigma^h / h$, argument of the Kanda roughness rescaling (–) |
The superscripts on $\lambda^p$ and $\lambda^f$ distinguish them from the longitude $\lambda$.
Two symbols here are shared with other land parameters and mean something different. $\beta$ is the drag correction of the Macdonald drag partition, not the moisture availability $\beta$ of the hydrology and interface-humidity models; it keeps the symbol of its source parametrization. Likewise $d$ is the zero-plane displacement, not the surface_thickness $d$ of SkinHumidity.
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) |
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.
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) |
Net surface freshwater fluxes
| 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) |
| $J^{\mathrm{ib}}$ | Jⁱᵇ | iceberg freshwater flux | Iceberg calving mass flux into the ocean (kg m⁻² s⁻¹) |
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) |
| $\mathcal{L}^f$ | ℒᶠ | latent heat of fusion | Heat the ocean supplies to melt snowfall and icebergs (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 $q = ε e / p$) (–) |
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) |
𝓃 | \scrn | Script n (pore-size uniformity) |
𝓂 | \scrm | Script m (van Genuchten 1 - 1/𝓃) |
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 |
ʰ | \^h | Superscript h |
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) |