Notation and conventions
This appendix establishes a common notation across the documentation and source code. Each entry lists a mathematical symbol and the Unicode form commonly used in the codebase, along with a common "property name", and a description. The property names may take a verbose "English form" or concise "mathematical form" corresponding to the given Unicode symbol. As properties, mathematical names are usually used mathematical form is invoked for the elements of a NamedTuple. Mathematical symbols are shown with inline math, while the Unicode column shows the exact glyphs used in code.
The table below reserves symbols for the dynamics and thermodynamics — the equations of motion, the thermodynamic state, and the numerics that step them — so that new notation in those can be introduced without colliding with what is already there. A parameterization dense enough to exhaust the alphabet on its own does not belong in it: its notation is scoped to its own pages instead, where reusing a letter reserved here is a local decision rather than a conflict. The P3 notation section does that for the Predicted Particle Properties scheme.
A few notes about the following table:
TCstands forThermodynamicConstantsAMstands forAtmosphereModelRSstands forReferenceState- Note that there are independent concepts of "reference". For example,
AnelasticDynamicsinvolves a "reference state", which is an adiabatic, hydrostatic solution to the equations of motion. But there is also an "energy reference temperature" and "reference latent heat", which are thermodynamic constants required to define the internal energy of moist atmospheric constituents. - Mapping to AM fields:
ρscorresponds tostatic_energy_density(model), and the moisture density is accessed viamodel.moisture_density.
The following table also uses a few conventions that suffuse the source code and which are internalized by wise developers:
constantsrefers to an instance ofThermodynamicConstants()qrefers to an instance ofMoistureMassFractions- "Reference" quantities use a subscript $r$ (e.g., $p_r$, $\rho_r$).
- A subscript plus marks the clamped form of a quantity, $x₊$: either $\max(0, x)$ or $x$ floored at the scheme's minimum (e.g. $nⁱ₊$, $qʳ₊$ in P3). The superscript $⁺$ is reserved for saturation ($qᵛ⁺$).
- Phase or mixture identifiers ($d$, $v$, $m$, and $t$ for total) appear as superscripts (e.g., $Rᵈ$, $cᵖᵐ$, $qᵗ$, $ρᵗ$), matching usage in the codebase (e.g.,
Rᵈ,cᵖᵐ). - The superscript $s$ is reserved for surface (e.g. $pˢ$, $θˢ$, $ρˢ$: values at a column's bottom face). Snow therefore takes $sn$ (e.g. $qˢⁿ$, $ρqˢⁿ$, $𝕎ˢⁿ$). Other multi-letter superscripts keep their own spellings: $st$ for standard ($pˢᵗ$), $sw$ for shortwave ($τˢʷ$), and $th$ for thermodynamic state ($ρᵗʰ$), distinct from $t$ for total.
- Momentum and the thermodynamic variable are stored continuity-density-weighted ($ρu = ρᵈ u$, and the thermodynamic density $ρᵡ = ρᵈ χ$ — i.e. $ρθ = ρᵈ θ$ or $ρs$). The continuity density is
continuity_density(dynamics): the reference density $ρᵣ$ forAnelasticDynamics, and the prognostic dry-air density $ρᵈ$ forCompressibleDynamics. Velocity and $θ$ are recovered by dividing by the continuity density. - Water/moisture is stored as partial densities (
ρqᵛ,ρqˡ,ρqⁱ, …; mass per volume) and recovered as mass fractions by dividing by the total air density $ρ = ρᵈ + ρᵗ$ ($qˣ = ρˣ/ρ$), where $ρᵗ = ρqᵛᵉ + Σ ρqᶜ$ is the total condensate density (all phases of the condensable species). The thermodynamics works in mass fractions throughout. The total density (total_density(dynamics)) is diagnosed on the compressible core; on the anelastic core it is the reference density $ρᵣ$, which the anelastic approximation substitutes for $ρ$. It also weights scalar and water advection, hydrostatics, and radiation. - Relations of the local thermodynamic state, such as the saturation specific humidity $qᵛ⁺ = pᵛ⁺ / (ρᵗʰ Rᵛ T)$ and the anelastic buoyancy, use the thermodynamic state density $ρᵗʰ$ (
thermodynamic_state_density, ordensity(𝒰, constants)for a state $𝒰$): the equation of state evaluated with the local temperature and moisture. It is $ρᵗʰ = pᵣ / (Rᵐ T)$ on the anelastic core, which differs from $ρᵣ$ wherever $T$ or $q$ depart from the reference state, and $ρᵗʰ = ρ$ on the compressible core. What distinguishes $ρᵗʰ$ is the state it is evaluated at, not its mathematical form: many densities satisfy some equation of state (the reference density $ρᵣ$ is the equation of state at the reference temperature and moisture), while $ρᵗʰ$ is the one at the local temperature and moisture.
| math symbol | code | property name | description |
|---|---|---|---|
| $\rho$ | ρ | AM.density | Total air density, $ρ = ρᵈ + ρᵗ$, total_density(dynamics): diagnosed on the compressible core; the reference density $ρᵣ$ on the anelastic core |
| $ρᵗʰ$ | ρᵗʰ | thermodynamic_state_density | Thermodynamic state density, the equation of state with the local $T$ and $q$: $ρᵗʰ = pᵣ / (Rᵐ T)$ on the anelastic core, $ρᵗʰ = ρ$ on the compressible core. Saturation is evaluated at $ρᵗʰ$, while $ρ$ carries mass and weights tendencies |
| $ρᵈ$ | ρᵈ | AM.dynamics.dry_density | Dry-air density; the prognostic continuity density of CompressibleDynamics (total $ρ = ρᵈ + ρᵗ$) |
| $ρᵗ$ | ρᵗ | Total condensate density, $ρᵗ = ρqᵛᵉ + Σ ρqᶜ$ (vapor/equilibrium moisture plus all condensed species); diagnosed via total_condensate_density, summed over condensate_field_names | |
| $\alpha$ | α | Specific volume, $α = 1/ρ$ | |
| $\boldsymbol{u} = (u,v,w)$ | u, v, w | AM.velocities | Velocity components in (x, y, z) or (east, north, up) |
| $\boldsymbol{ρu} = (ρu, ρv, ρw)$ | ρu, ρv, ρw | AM.momentum | Momentum components |
| $\tilde{w}$ | w̃ | AM.dynamics.contravariant_vertical_velocity | Contravariant vertical velocity (grid-relative, normal to $r$-surfaces) |
| $\rho \tilde{w}$ | ρw̃ | AM.dynamics.contravariant_vertical_momentum | Contravariant vertical momentum |
| $r$ | r | rnode(i, j, k, grid, ℓz) | Reference (computational) vertical coordinate of a terrain-following grid; the physical height is $z(x, y, r)$ (znode), matching Oceananigans' r/z convention |
| $s$ | s | static_energy(model) | (Liquid-ice) moist static energy, $s = cᵖᵐ T + g z - ℒˡ₀ qˡ - ℒⁱ₀ qⁱ$; $e$ is reserved for turbulent kinetic energy |
| $ρ s$ | ρs | static_energy_density(model) | Static energy density, the prognostic thermodynamic variable of StaticEnergyFormulation |
| $E$ | E | total_energy(model) | Total energy, $E = s + (u^2 + v^2 + w^2)/2$. E is also the formulation-agnostic energy key: a flux or forcing supplied under E (or ρE) is applied to whichever thermodynamic variable the formulation evolves |
| $ρ E$ | ρE | Total energy density; the boundary_conditions and forcing key for an energy input, routed onto $ρᵡ$ | |
| $ρᵡ$ | ρᵡ | thermodynamic_density(formulation) | Thermodynamic density: the generic continuity-density-weighted prognostic thermodynamic variable, $ρᵡ = ρᵈ χ$ — concretely $ρθ$ for the potential-temperature formulation or $ρs$ for static energy. The intensive variable is recovered as $χ = ρᵡ / ρᵈ$ |
| $T$ | T | AM.temperature | Temperature |
| $T⁺$ | T⁺ | DewpointTemperature(model) | Dewpoint temperature |
| $p$ | p | AM.pressure | Pressure |
| $b$ | b | Buoyancy | |
| $q^{ve}$ | qᵛᵉ | Scheme-dependent specific moisture: vapor (non-equilibrium) or equilibrium moisture (saturation adjustment) | |
| $ρ q^{ve}$ | ρqᵛᵉ | AM.moisture_density | Scheme-dependent moisture density: $ρqᵛ$ or $ρqᵉ$ |
| $ρ qᵉ$ | ρqᵉ | AM.moisture_density | Equilibrium moisture density (saturation adjustment schemes) |
| $ρ qᵛ$ | ρqᵛ | AM.moisture_density | Vapor density (non-equilibrium schemes) |
| $ρ qᵗ$ | ρqᵗ | Total moisture density; the boundary_conditions and forcing key for a water input (specific alias qᵗ for forcings), routed onto $ρqᵛᵉ$ whatever the microphysics calls it | |
| $qᵛ$ | qᵛ | AM.microphysical_fields.qᵛ | Vapor mass fraction, a.k.a "specific humidity" |
| $qᵛ⁺ˡ$ | qᵛ⁺ˡ | Saturation vapor mass fraction over liquid water (kg/kg) | |
| $δqᵛ⁺ˡ$ | δqᵛ⁺ˡ | AM.microphysical_fields.δqᵛ⁺ˡ | Vapor excess over liquid, $δqᵛ⁺ˡ = qᵛ − qᵛ⁺ˡ$ (kg/kg); P3's optional prognostic |
| $δqᵛ⁺ⁱ$ | δqᵛ⁺ⁱ | Vapor excess over ice, $δqᵛ⁺ⁱ = qᵛ − qᵛ⁺ⁱ$ (kg/kg) | |
| $ρδqᵛ⁺ˡ$ | ρδqᵛ⁺ˡ | AM.microphysical_fields.ρδqᵛ⁺ˡ | Density-weighted vapor excess over liquid (kg/m³); may be negative |
| $qˡ$ | qˡ | AM.microphysical_fields.qˡ | Liquid mass fraction |
| $qⁱ$ | qⁱ | AM.microphysical_fields.qⁱ | Ice mass fraction |
| $qᶜˡ$ | qᶜˡ | AM.microphysical_fields.qᶜˡ | Cloud liquid mass fraction |
| $qᶜⁱ$ | qᶜⁱ | AM.microphysical_fields.qᶜⁱ | Cloud ice mass fraction |
| $q^f$ | qᶠ | AM.microphysical_fields.qᶠ | Rime mass fraction (P3); a portion of $qⁱ$, not additional mass |
| $b^f$ | bᶠ | AM.microphysical_fields.bᶠ | Rime volume per unit mass (P3), m³/kg |
| $q^{wi}$ | qʷⁱ | AM.microphysical_fields.qʷⁱ | Liquid mass fraction carried on ice particles (P3); counted in $qˡ$, but sediments with the ice |
| $qʳ$ | qʳ | Rain mass fraction | |
| $qˢⁿ$ | qˢⁿ | Snow mass fraction | |
| $\hat{q}^x$ | q̂ˣ | Unit composition: mass fractions with all mass in constituent $x$ ($\hat{q}^d$ is dry air) | |
| $Δq$ | Δq | Composition increment (a difference of mass-fraction vectors), e.g. per unit sedimenting mass | |
| $ρqᵛ$ | ρqᵛ | Vapor density | |
| $ρqˡ$ | ρqˡ | Liquid density | |
| $ρqⁱ$ | ρqⁱ | Ice density | |
| $ρqᶜˡ$ | ρqᶜˡ | Cloud liquid density | |
| $ρqᶜⁱ$ | ρqᶜⁱ | Cloud ice density | |
| $ρqʳ$ | ρqʳ | AM.microphysical_fields.ρqʳ | Rain density |
| $ρqˢⁿ$ | ρqˢⁿ | AM.microphysical_fields.ρqˢⁿ | Snow density |
| $n^{cl}$ | nᶜˡ | AM.microphysical_fields.nᶜˡ | Cloud droplet number per unit mass (1/kg) |
| $n^r$ | nʳ | AM.microphysical_fields.nʳ | Rain drop number per unit mass (1/kg) |
| $n^a$ | nᵃ | AM.microphysical_fields.nᵃ | Aerosol number per unit mass (1/kg) |
| $\rho n^{cl}$ | ρnᶜˡ | AM.microphysical_fields.ρnᶜˡ | Cloud droplet number density (1/m³) |
| $\rho n^r$ | ρnʳ | AM.microphysical_fields.ρnʳ | Rain drop number density (1/m³) |
| $\rho n^a$ | ρnᵃ | AM.microphysical_fields.ρnᵃ | Aerosol number density (1/m³) |
| $N^{cl}$ | Nᶜˡ | Volumetric cloud droplet number density, $Nᶜˡ = ρ nᶜˡ$ (1/m³) | |
| $N^r$ | Nʳ | Volumetric rain drop number density, $Nʳ = ρ nʳ$ (1/m³) | |
| $N^a$ | Nᵃ | Volumetric aerosol number density, $Nᵃ = ρ nᵃ$ (1/m³) | |
| $n^{ccn}$ | nᶜᶜⁿ | Cloud condensation nuclei activated per unit mass (1/kg); the activation rate $∂nᶜᶜⁿ/∂t$ is a source of $nᶜˡ$ | |
| $N^{ccn}$ | Nᶜᶜⁿ | Volumetric CCN number density, $Nᶜᶜⁿ = ρ nᶜᶜⁿ$ (1/m³) | |
| $q^{ccn}$ | qᶜᶜⁿ | Mass fraction condensed onto newly activated CCN (kg/kg) | |
| $\mathbb{W}^{cl}$ | 𝕎ᶜˡ | Terminal velocity of cloud liquid (scalar, positive downward) | |
| $\mathbb{W}^{ci}$ | 𝕎ᶜⁱ | Terminal velocity of cloud ice (scalar, positive downward) | |
| $\mathbb{W}^r$ | 𝕎ʳ | Terminal velocity of rain (scalar, positive downward) | |
| $\mathbb{W}^{sn}$ | 𝕎ˢⁿ | Terminal velocity of snow (scalar, positive downward) | |
| $\mathbb{W}^i$ | 𝕎ⁱ | Terminal velocity of ice (scalar, positive downward); $i$ is dry ice, distinct from cloud ice $ci$ | |
| $\mathbb{W}^{nx}$ | 𝕎ⁿˣ | Number-weighted terminal velocity of species $x$ (𝕎ⁿᶜˡ, 𝕎ⁿʳ, 𝕎ⁿⁱ); a bare species label is the mass-weighted mean, and the weighting marker is a superscript preceding the species | |
| $w^x$ | wˣ | Signed vertical advection velocity of species $x$, $wˣ = -𝕎ˣ$ (wᶜˡ, wⁿᶜˡ, wʳ, wⁿʳ, wⁱ, wⁿⁱ) | |
| $qᵛ⁺$ | qᵛ⁺ | Saturation specific humidity over a surface | |
| $qᵛ⁺ˡ$ | qᵛ⁺ˡ | Saturation specific humidity over a planar liquid surface | |
| $qᵛ⁺ⁱ$ | qᵛ⁺ⁱ | Saturation specific humidity over a planar ice surface | |
| $pᵛ$ | pᵛ | Vapor pressure (partial pressure of water vapor), $pᵛ = ρ qᵛ Rᵛ T$ | |
| $pᵛ⁺$ | pᵛ⁺ | Saturation vapor pressure | |
| $\mathscr{H}$ | ℋ | RelativeHumidity(model) | Relative humidity, $ℋ = pᵛ / pᵛ⁺$ |
| $\mathscr{S}$ | 𝒮 | supersaturation(T, ρ, q, c, surf), Supersaturation(model) | Supersaturation, $𝒮 = pᵛ / pᵛ⁺ - 1$ |
| $ξ$ | ξ | psychrometric_correction | Psychrometric correction, $ξ = 1 + ℒ² qᵛ⁺ / (cᵖ Rᵛ T²)$; $ξˡ$ and $ξⁱ$ name the liquid and ice phase |
| $δqˡ$, $δqⁱ$ | δqˡ, δqⁱ | Saturation-adjustment increments, $δq = (qᵛ - qᵛ⁺) / ξ$ for the liquid and ice phase | |
| $g$ | g | TC.gravitational_acceleration | Gravitational acceleration |
| $c^{ac}$ | cᵃᶜ | Acoustic sound speed, $cᵃᶜ = \sqrt{γ Rᵈ T}$ | |
| $\mathbb{C}_{X,i}$ | ℂˣᵢ | descriptive parameter property | The $i$-th calibratable empirical coefficient in relation $X$; source uses modifier letters because Unicode lacks general subscript letters; state, physical constants, case inputs, switches, and numerical safeguards do not receive $\mathbb{C}$ |
| $\mathcal{R}$ | ℛ | TC.molar_gas_constant | Universal (molar) gas constant |
| $Tᵗʳ$ | Tᵗʳ | TC.triple_point_temperature | Temperature at the vapor-liquid-ice triple point |
| $pᵗʳ$ | pᵗʳ | TC.triple_point_pressure | Pressure at the vapor-liquid-ice triple point |
| $mᵈ$ | mᵈ | TC.dry_air.molar_mass | Molar mass of dry air |
| $mᵛ$ | mᵛ | TC.vapor.molar_mass | Molar mass of vapor |
| $Rᵈ$ | Rᵈ | dry_air_gas_constant(constants) | Dry air gas constant ($Rᵈ = \mathcal{R} / mᵈ$) |
| $Rᵛ$ | Rᵛ | vapor_gas_constant(constants) | Water vapor gas constant ($Rᵛ = \mathcal{R} / mᵛ$) |
| $Rᵐ$ | Rᵐ | mixture_gas_constant(q, constants) | Mixture gas constant, function of $q$ |
| $cᵖᵈ$ | cᵖᵈ | TC.dry_air.heat_capacity | Heat capacity of dry air at constant pressure |
| $cᵖᵛ$ | cᵖᵛ | TC.vapor.heat_capacity | Heat capacity of vapor at constant pressure |
| $cˡ$ | cˡ | TC.liquid.heat_capacity | Heat capacity of the liquid phase (incompressible) |
| $cⁱ$ | cⁱ | TC.ice.heat_capacity | Heat capacity of the ice phase (incompressible) |
| $\rho^L$ | ρᴸ | TC.liquid.density | Intrinsic density of liquid water |
| $\rho^I$ | ρᴵ | TC.ice.density | Intrinsic density of ice |
| $cᵖᵐ$ | cᵖᵐ | mixture_heat_capacity(q, constants) | Mixture heat capacity at constant pressure |
| $cᵛᵐ$ | cᵛᵐ | Mixture heat capacity at constant volume, $cᵛᵐ = cᵖᵐ - Rᵐ$ | |
| $\hat{c}ᵖ$ | ĉᵖ | vapor_equivalent_heat_capacity(q, constants) | Vapor-equivalent heat capacity, $ĉᵖ = qᵈ cᵖᵈ + qᵗ cᵖᵛ$: the heat capacity if all water were vapor; unchanged by phase change |
| $\hat{R}$ | R̂ | vapor_equivalent_gas_constant(q, constants) | Vapor-equivalent gas constant, $R̂ = qᵈ Rᵈ + qᵗ Rᵛ$; equals $Rᵐ$ in unsaturated air |
| $\hat{κ}$ | κ̂ | exner_exponent(VaporEquivalentExponent(), q, constants) | Vapor-equivalent Exner exponent, $κ̂ = R̂ / ĉᵖ$ (the default exponent of $θˡⁱ$); $κᵈ = Rᵈ / cᵖᵈ$ is the DryExponent. In code, κ̂ also names the exponent choice (VaporEquivalentExponent() or DryExponent()) passed to functions. Distinct from the von Kármán constant $κ$ |
| $Tᵣ$ | Tᵣ | TC.energy_reference_temperature | Reference temperature for internal energy relations and latent heat |
| $\mathcal{L}^l_r$ | ℒˡᵣ | TC.liquid.reference_latent_heat | Latent heat of condensation at the energy reference temperature |
| $\mathcal{L}^i_r$ | ℒⁱᵣ | TC.ice.reference_latent_heat | Latent heat of deposition at the energy reference temperature |
| $ℒˡ₀, ℒⁱ₀$ | ℒˡ₀, ℒⁱ₀ | absolute_zero_latent_heat(TC, phase) | Latent-heat intercepts at zero kelvin, used with absolute-temperature sensible energy |
| $H^c$ | Hᶜ | condensate_latent_energy(q, TC) | Latent energy of the condensate, $Hᶜ = ℒˡ₀ qˡ + ℒⁱ₀ qⁱ$, so that $H = cᵖᵐ T - Hᶜ$ |
| $\mathcal{L}^l(T)$ | ℒˡ | liquid_latent_heat(T, constants) | Temperature-dependent latent heat of condensation |
| $\mathcal{L}^i(T)$ | ℒⁱ | ice_latent_heat(T, constants) | Temperature-dependent latent heat of deposition |
| $\mathcal{L}^{fr}(T)$ | ℒᶠʳ | fusion_latent_heat(constants, T) | Temperature-dependent latent heat of fusion (freezing), $ℒᶠʳ = ℒⁱ - ℒˡ$; the superscript fr abbreviates freezing, since f labels rime (qᶠ, bᶠ) |
| $θ₀$ | θ₀ | RS.potential_temperature | (Constant) reference potential temperature for the anelastic formulation |
| $p₀$ | p₀ | RS.base_pressure | Reference pressure at $z = 0$: the datum the reference profiles are anchored to |
| $pˢ$ | pˢ | RS.surface_pressure | Reference pressure at a column's bottom face (the terrain surface over terrain), $p₀$ reduced to that height |
| $p^{st}$ | pˢᵗ | RS.standard_pressure | Standard pressure for potential temperature (default 10⁵ Pa) |
| $ρᵣ$ | ρᵣ | RS.density | Density of a dry reference state for the anelastic formulation |
| $αᵣ$ | αᵣ | Specific volume of a dry reference state, $αᵣ = Rᵈ θ₀ / pᵣ$ | |
| $pᵣ$ | pᵣ | RS.pressure | Pressure of a dry adiabatic reference pressure for the anelastic formulation |
| $\Pi$ | Π | exner_function(𝒰, constants) | Exner function of the liquid-ice potential temperature, $Π = (p / pˢᵗ)^κ$, with $κ = κ̂$ (VaporEquivalentExponent, default) or $κᵈ$ (DryExponent); $p = pᵣ$ on the anelastic core |
| $\hat{Π}$ | Π̂ | Vapor-equivalent Exner function, $Π̂ = (p / pˢᵗ)^{κ̂}$ | |
| $θᵛ$ | θᵛ | Virtual potential temperature | |
| $θᵉ$ | θᵉ | Equivalent potential temperature | |
| $θˡⁱ$ | θˡⁱ | LiquidIcePotentialTemperature(model) | Liquid-ice potential temperature, $θˡⁱ = H / (ĉᵖ Π)$; $θˡⁱ = T / Π$ in unsaturated air |
| $H$ | H | Liquid-ice enthalpy, $H = cᵖᵐ T - ℒˡ₀ qˡ - ℒⁱ₀ qⁱ = ĉᵖ Π θˡⁱ$; invariant under phase change at fixed $p$ and $qᵗ$ | |
| $θᵇ$ | θᵇ | Stability-equivalent potential temperature (for moist Brunt-Väisälä) | |
| $θ$ | θ | Shorthand for liquid-ice potential temperature (used in set!) | |
| $\Delta t$ | Δt | Simulation.Δt | Time step. |
| $\boldsymbol{\tau}$ | τ | Kinematic subgrid/viscous stress tensor (per unit mass) | |
| $\boldsymbol{\mathcal{T}}$ | 𝒯 | Dynamic stress tensor used in anelastic momentum, $\mathcal{T} = ρᵣ τ$ | |
| $\boldsymbol{J}$ | J | Dynamic diffusive flux for scalars | |
| $τˣ$ | τˣ | Surface momentum flux ($x$-component), N/m² | |
| $τʸ$ | τʸ | Surface momentum flux ($y$-component), N/m² | |
| $\mathcal{Q}^T$ | 𝒬ᵀ | Surface sensible heat flux, $\mathcal{Q}^T = cᵖᵐ Jᵀ$ | |
| $\mathcal{Q}^v$ | 𝒬ᵛ | Surface latent heat flux, $\mathcal{Q}^v = \mathcal{L}^l Jᵛ$ | |
| $Jᵀ$ | Jᵀ | Surface temperature flux, kg K/m²s | |
| $Jᶿ$ | Jᶿ | Surface potential-temperature flux, $Jᶿ = 𝒬ᵀ / (ĉᵖ Π)$, kg K/m²s | |
| $Jᵛ$ | Jᵛ | Surface moisture flux, kg/m²s | |
| $Cᴰ$ | Cᴰ | Surface drag coefficient | |
| $Cᵀ$ | Cᵀ | Surface sensible heat transfer coefficient (Stanton number) | |
| $Cᵛ$ | Cᵛ | Surface vapor transfer coefficient (Dalton number) | |
| $Uᵍ$ | Uᵍ | gustiness | Gustiness: the unresolved wind added to the resolved wind in the bulk surface fluxes, m/s; $Uᵍ = \max(c \sqrt{e}, Uᵍ_{min})$ for TKEGustiness |
| $Ũ$ | Ũ | Effective wind speed of the bulk surface fluxes and their transfer coefficients, $Ũ = \sqrt{u² + v² + Uᵍ²}$, m/s | |
| $\ell$ | ℓ | A length scale, m; the superscript says which one. Bare $ℓ$ is used locally where only one length scale is in play (the primary mixing length in TKEBasedTurbulenceClosure, the divergence-damping scale in CompressibleEquations) | |
| $\ell^b$ | ℓᵇ | Buoyancy penetration depth, $ℓᵇ = \sqrt{e} / N$, how far an eddy with kinetic energy $e$ penetrates air of buoyancy frequency $N$; infinite where $N² ≤ 0$. Every mixing-length formulation is built from it and the wall length $ℂˢ z$: the local minimum $\min(ℂˢ z, ℓᵇ)$, its interior envelope with slope $ℂⁱ$, capped separately by $ℂˢ z$ (the default), or the Bougeault–Lacarrère parcel lengths | |
| $\ell^u, \ell^c, \ell^e, \ell^D$ | ℓᵘ, ℓᶜ, ℓᵉ, ℓᴰ | Mixing lengths for momentum, scalars, TKE and dissipation: $ℓᵘ = Sᵘ ℓ$, $ℓᶜ = Sᶜ ℓ$, $ℓᵉ = Sᵉ ℓ$, $ℓᴰ = ℓ / Sᴰ$ | |
| $S^u, S^c, S^e, S^D$ | Sᵘ, Sᶜ, Sᵉ, Sᴰ | Stability functions of TKEBasedTurbulenceClosure; constants $ℂᵘ, ℂᶜ, ℂᵉ, ℂᴰ$ in ConstantStabilityFunctions | |
| $\ell^r$ | ℓʳ | roughness_length | Surface roughness length for momentum, m. Superscript $r$ elsewhere denotes rain ($qʳ$); the two never appear together |
| $\ell^{rh}$ | ℓʳʰ | scalar_roughness_length | Surface roughness length for heat and moisture; defaults to $ℓʳ/7.3$ |
| $\kappa$ | κ | von Kármán constant. Never used for a diffusivity in Breeze's own code — the Oceananigans accessors literally named κᶠᶜᶜ, κᶜᶠᶜ, κᶜᶜᶠ are the one exception, and they are thin one-liners | |
| $K^u$ | Kᵘ | Eddy diffusivity for momentum, $K^u = ℓ^u \sqrt{e}$, m² s⁻¹. Oceananigans' scalar-diffusivity closures spell their own field νₑ, which Breeze reads but does not define | |
| $K^c$ | Kᶜ | Eddy diffusivity for scalars, $K^c = ℓ^c \sqrt{e}$, m² s⁻¹ | |
| $K^e$ | Kᵉ | Eddy diffusivity for TKE, $K^e = ℓ^e \sqrt{e}$, m² s⁻¹ | |
| $e$ | e | Subgrid turbulent kinetic energy, m² s⁻² | |
| $eᵐⁱⁿ$ | eᵐⁱⁿ | minimum_tke | Floor on $e$ inside $\sqrt{e}$ of the diffusivities and length scales, and of the buoyancy sink rate $ωᴮ = -B⁻ / \max(e, eᵐⁱⁿ)$, m² s⁻² |
| $ωᴮ$ | ωᴮ | Implicit buoyancy sink rate of the TKE equation, $-B⁻ / \max(e, eᵐⁱⁿ)$ with $B⁻$ the negative part of the buoyancy flux, s⁻¹ | |
| $\rho e$ | ρe | Prognostic TKE density, the tracer TKEBasedTurbulenceClosure adds to the model | |
| $\mathrm{Pr}$ | Pr | Turbulent Prandtl number, $\mathrm{Pr} = K^u/K^c = S^u/S^c$ | |
| $Ri$ | Ri | Gradient Richardson number, $Ri = N²/S²$; the rational stability functions use $N² / (S² + S²ᵐⁱⁿ)$ | |
| $ℂˢ, ℂⁱ$ | ℂˢ, ℂⁱ | Wall-length coefficient and independent interior envelope slope. Coefficients of TKEBasedTurbulenceClosure are written $ℂ$, reserving $C$ for bulk transfer coefficients such as the surface drag $Cᴰ$ | |
| $ℓᵐ$ | ℓᵐ | Stored turbulence mixing length (closure_fields.ℓᵐ) | |
| $ℓˢ$ | ℓˢ | Stratification-only envelope before capping at the surface length | |
| $Ri⁻, Ri⁺$ | Ri⁻, Ri⁺ | Unstable and stable transition scales of the rational stability functions | |
| $S²ᵐⁱⁿ$ | S²ᵐⁱⁿ | minimum_shear_squared | Shear floor of the Richardson number of the rational stability functions, $Ri = N² / (S² + S²ᵐⁱⁿ)$, s⁻² |
| $S$ | S | Vertical shear magnitude, $S² = (∂_z u)² + (∂_z v)²$, s⁻¹ | |
| $N^2$ | N² | Squared Brunt–Väisälä frequency, s⁻². (Bare $N$ is the acoustic substep count, below) | |
| $h^{bl}$ | hᵇˡ | Boundary-layer depth, m; a diagnostic (the height of the capping inversion in convective conditions) | |
| $T_0$ | T₀ | Sea surface temperature | |
| $qᵛ₀$ | qᵛ₀ | Saturation specific humidity at sea surface | |
| $\mathscr{I}$ | ℐ | Radiative flux (intensity), W/m² | |
| $F_{\mathscr{I}}$ | Fℐ | Radiative flux divergence (heating rate), K/s | |
| $τˡʷ$ | τˡʷ | Atmosphere optical thickness for longwave | |
| $τˢʷ$ | τˢʷ | Atmosphere optical thickness for shortwave | |
| $N_A$ | ℕᴬ | Avogadro's number, molecules per mole | |
| $\mathcal{U}$ | 𝒰 | Thermodynamic state struct (e.g., StaticEnergyState) | |
| $\mathcal{M}$ | ℳ | Microphysical state struct (e.g., WarmPhaseOneMomentState) | |
| $θ^L$ | θᴸ | AcousticSubstepper.linearization_potential_temperature | Stage-entry potential temperature, $θ^L = (ρθ)^L / ρ^L$ |
| $C^L$ | Cᴸ | AcousticSubstepper.linearization_pressure_coefficient | Linearized pressure coefficient, the slope $∂p/∂(ρθ)$ of the equation of state at the stage-entry state, $C^L = γ^e p^L / (ρθ)^L$ |
| $ζ$ | ζ | adjusted_partition_response | Ratio of the slope $∂p/∂(ρθ)$ of the saturation-adjusted pressure to the slope at the held partition, at fixed densities |
| $γ^e$ | γᵉ | Effective heat-capacity ratio of the linearized equation of state, $γ^e = (θ / T) ∂T/∂θ$ | |
| $θ ∂T/∂θ$ | θ∂T∂θ | temperature_and_sensitivity(𝒰, constants) | Temperature sensitivity at fixed density and composition, propagated through the selected inversion |
| $\tilde{T}$ | T̃ | Condensate-free root of the density-based inversion of $θˡⁱ$ (the root with $H^c = 0$) | |
| $G^n$ | Gⁿ | Tendency fields at time step $n$ | |
| $G^s$ | Gˢ | Slow tendencies (excludes fast pressure gradient and buoyancy) | |
| $N$ | N | AcousticSubstepper.substeps | Acoustic substeps per outer time step, or adaptive substep count before stage partitioning |
| $N_τ$ | Nτ | Acoustic substeps in one Runge-Kutta stage | |
| $\Delta \tau$ | Δτ | Acoustic substep size, $Δτ = Δt / N$ for proportional substep distribution | |
| $ρ'$ | ρ′ | AcousticSubstepper.density_perturbation | Density perturbation about the stage-entry state |
| $(ρθ)'$ | ρθ′ | AcousticSubstepper.density_potential_temperature_perturbation | Thermodynamic-density perturbation about the stage-entry state |
| $(ρu)', (ρv)', (ρw)'$ | ρu′, ρv′, ρw′ | AcousticSubstepper.momentum_perturbation | Momentum perturbations about the stage-entry state |
| $D_τ$ | Dτ | Klemp-Skamarock-Ha divergence-damping proxy, $D_τ = ((ρθ)'_τ - (ρθ)'_{τ-\Delta τ}) / θ^L$ | |
| $\bar{u}, \bar{v}, \bar{w}$ | ū, v̄, w̄ | AcousticSubstepper.time_averaged_velocities | Time-averaged velocities for non-acoustic scalar advection |