HoltslagBovilleNonlocalABL
Nonlocal atmospheric boundary-layer diffusion scheme of Holtslag & Boville (1993). Given the height z, boundary-layer depth h, friction velocity u_star, surface virtual heat flux wθ_v0, surface scalar flux wC_0, surface virtual potential temperature θ_v0 and Obukhov length L, it computes the convective velocity scale w_star, the φ_m/φ_h Monin-Obukhov similarity profiles, the momentum and scalar velocity scales w_m/w_t, the turbulent Prandtl number Pr, the eddy-diffusivity profile Kc = κ·w_t·z·(1−z/h)² and the countergradient nonlocal transport term γc. The nonlocal scheme represents transport by large convective eddies that the local scheme cannot. Migrated from AtmosphericDynamics.jl@816cc391 src/holtslag_boville_1993.jl:290 (`HoltslagBovilleNonlocalABL` @component); algebraic (observed-only) — the 7 inputs are parameters, the 14 turbulence quantities are algebraic state variables, no ODE integration.
Description
Nonlocal atmospheric boundary-layer diffusion scheme of Holtslag & Boville (1993). Given the height z, boundary-layer depth h, friction velocity u_star, surface virtual heat flux wθ_v0, surface scalar flux wC_0, surface virtual potential temperature θ_v0 and Obukhov length L, it computes the convective velocity scale w_star, the φ_m/φ_h Monin-Obukhov similarity profiles, the momentum and scalar velocity scales w_m/w_t, the turbulent Prandtl number Pr, the eddy-diffusivity profile Kc = κ·w_t·z·(1−z/h)² and the countergradient nonlocal transport term γc. The nonlocal scheme represents transport by large convective eddies that the local scheme cannot. Migrated from AtmosphericDynamics.jl@816cc391 src/holtslag_boville_1993.jl:290 (HoltslagBovilleNonlocalABL @component); algebraic (observed-only) — the 7 inputs are parameters, the 14 turbulence quantities are algebraic state variables, no ODE integration.
Reference
Holtslag & Boville, 1993
source_ref: AtmosphericDynamics.jl@816cc3918e1eef2c2ee6d8fa8dd73c8f13a85a47:src/holtslag_boville_1993.jl:290
Upstream HoltslagBovilleNonlocalABL is an MTK @component that returns System(eqs, t) with 14 time-indexed @variables, each fixed by one algebraic equation (no time derivative appears), and 7 @parameters of inputs. It is migrated as an algebraic (observed-only) model: the 7 inputs are parameters, the 14 turbulence quantities are state variables defined by the algebraic equations — the same serialization pattern as components/atmospheric_dynamics/sp_ch1/ideal_gas.esm and components/wildland_fire/rothermel/wind_limit.esm. No ODE integration occurs; tests assert the closed-form solution at t=0.
The 11 upstream @constants (g, κ, a_coeff, c_1, ε_sl, w_ref, zero_heat_flux, zero_length, γc_zero, zero_velocity, L_min) are preserved as parameter declarations with their upstream defaults, matching the MTK-@constants-to-ESM-parameter convention used in wind_limit.esm. The zero_heat_flux/zero_length/zero_velocity/γc_zero constants are kept rather than inlined as 0.0 literals so the ifelse comparison operands and stable-branch values carry units matching the quantities they are compared with (K*m/s, m, m/s, 1/m).
Variable names: Unicode subscripts in the upstream identifiers are transliterated to ASCII (φₕ→φ_h, φₘ→φ_m, φₕ_local→φ_h_local, φₘ_local→φ_m_local, wₘ→w_m, wₘ_outer→w_m_outer, wₜ→w_t, wθᵥ₀→wθ_v0, wC₀→wC_0, θᵥ₀→θ_v0, c₁→c_1); Greek base letters (φ, ζ, η, κ, ε, γ, θ) are retained, following the convention in components/urban_canopy/urban_radiation.esm.
Regularization: ζ = z/L and ζ_ε = ε_sl·h/L are guarded for small |L| — when abs(L) ≤ L_min the denominator L is replaced by a value of magnitude L_min carrying L’s sign. Upstream encodes this branchlessly as L_min·sign(L) + L_min·(1−abs(sign(L))); it is re-expressed here as the mathematically identical ifelse(L < 0, −L_min, L_min) because the ESS Julia MTK binding (EarthSciASTMTKExt.jl) does not implement the sign operator that esm-spec §4.2 lists — the two forms agree exactly for L>0 (+L_min), L<0 (−L_min) and L=0 (+L_min). The regularized branch is never selected by the inline tests (all use |L| ≥ 50 m); it only needs to compile and stay finite.
Tolerance rationale (model-level rel=1e-6): observed-only algebraic system (no ODE integration); every output is a closed-form composition of arithmetic, sqrt, cube-root (^1/3) and ifelse selections in float64. 1e-6 is far above float64 round-off accumulated over the ~10-deep expression chains (w_t depends on w_m and Pr, which depend on the φ profiles and w_star) and below any physical-discrimination scale on the m/s, m^2/s and 1/m output quantities. Exact-zero expectations (w_star and γc in stable conditions, where the ifelse selects the zero-velocity / γc_zero constant) carry a per-assertion abs=1e-12 tolerance.
- DOI: 10.1175/1520-0442(1993)006<1825:LVNBLD>2.0.CO;2
Parameters
| Name | Units | Default | Description |
|---|---|---|---|
z | m | 100.0 | Height above surface |
h | m | 1000.0 | Boundary layer height |
u_star | m/s | 0.3 | Friction velocity |
wθ_v0 | K*m/s | 0.05 | Surface virtual heat flux (w’θ_v’)_0 |
wC_0 | m/s | 1e-05 | Surface kinematic flux of scalar C (e.g. mixing ratio) |
θ_v0 | K | 300.0 | Virtual potential temperature at surface |
L | m | -100.0 | Obukhov length (negative for unstable, positive for stable) |
g | m/s^2 | 9.81 | Gravitational acceleration |
κ | 1 | 0.4 | von Karman constant |
a_coeff | 1 | 7.2 | Nonlocal transport constant (Eq. A14: a ≈ 7.2) |
c_1 | 1 | 0.6 | Velocity scale constant (Eq. A11) |
ε_sl | 1 | 0.1 | Surface layer fraction of boundary layer height |
w_ref | m/s | 1.0 | Reference velocity scale for non-dimensionalizing the cube-root arguments |
zero_heat_flux | K*m/s | 0.0 | Zero heat flux, comparison constant for the unstable/stable branch |
zero_length | m | 0.0 | Zero length, comparison constant for the L sign branch |
γc_zero | 1/m | 0.0 | Zero nonlocal transport term (stable-branch value of γc) |
zero_velocity | m/s | 0.0 | Zero velocity (stable-branch value of w_star) |
L_min | m | 1.0 | Minimum absolute Obukhov length for regularization of z/L |
Observed
| Name | Units | Default | Description |
|---|---|---|---|
ζ | 1 | Stability parameter z/L at local height | |
η | 1 | Relative height z/h | |
ζ_ε | 1 | Stability parameter ε_sl·h/L at the surface layer top | |
w_star | m/s | Convective velocity scale (Eq. A12) | |
φ_m | 1 | Dimensionless wind shear at the surface layer top (Eq. A2/A4/A8) | |
φ_h | 1 | Dimensionless temperature gradient at the surface layer top (Eq. A3/A5/A6) | |
φ_h_local | 1 | Dimensionless temperature gradient at local height z (Eq. A3/A5/A6) | |
φ_m_local | 1 | Dimensionless wind shear at local height z (Eq. A2/A4/A8) | |
w_m_outer | m/s | Outer-layer momentum velocity scale (Eq. A11) | |
w_m | m/s | Velocity scale for momentum, height-dependent (Eq. A7/A11) | |
Pr | 1 | Turbulent Prandtl number (Eq. A13) | |
w_t | m/s | Velocity scale for scalars, height-dependent (Eq. A1/A10) | |
Kc | m^2/s | Eddy diffusivity profile (Eq. 3.9) | |
γc | 1/m | Nonlocal countergradient transport term (Eq. 3.10) |
Observed expressions
$$ ζ = \mathrm{ifelse}\left(\mathrm{>}\left(\left|L\right|, L_{min}\right), \frac{z}{L}, \frac{z}{\mathrm{ifelse}\left(\mathrm{<}\left(L, zero_{length}\right), -L_{min}, L_{min}\right)}\right) $$Stability parameter z/L at local height
$$ η = \frac{z}{h} $$Relative height z/h
$$ ζ_{ε} = \mathrm{ifelse}\left(\mathrm{>}\left(\left|L\right|, L_{min}\right), \frac{ε_{sl} \cdot h}{L}, \frac{ε_{sl} \cdot h}{\mathrm{ifelse}\left(\mathrm{<}\left(L, zero_{length}\right), -L_{min}, L_{min}\right)}\right) $$Stability parameter ε_sl·h/L at the surface layer top
$$ w_{star} = \mathrm{ifelse}\left(\mathrm{>}\left(wθ_{v0}, zero_{heat\,flux}\right), \frac{\frac{g}{θ_{v0}} \cdot wθ_{v0} \cdot h}{w_{ref}^{3}}^{\frac{1.0}{3.0}} \cdot w_{ref}, zero_{velocity}\right) $$Convective velocity scale (Eq. A12)
$$ φ_{m} = \mathrm{ifelse}\left(\mathrm{>}\left(L, zero_{length}\right), \mathrm{ifelse}\left(\mathrm{<=}\left(ζ_{ε}, 1\right), 1 + 5 \cdot ζ_{ε}, 5 + ζ_{ε}\right), \mathrm{ifelse}\left(\mathrm{<}\left(L, zero_{length}\right), \frac{1}{\left(1 - 15 \cdot ζ_{ε}\right)^{\frac{1.0}{3.0}}}, 1.0\right)\right) $$Dimensionless wind shear at the surface layer top (Eq. A2/A4/A8)
$$ φ_{h} = \mathrm{ifelse}\left(\mathrm{>}\left(L, zero_{length}\right), \mathrm{ifelse}\left(\mathrm{<=}\left(ζ_{ε}, 1\right), 1 + 5 \cdot ζ_{ε}, 5 + ζ_{ε}\right), \mathrm{ifelse}\left(\mathrm{<}\left(L, zero_{length}\right), \frac{1}{\sqrt{1 - 15 \cdot ζ_{ε}}}, 1.0\right)\right) $$Dimensionless temperature gradient at the surface layer top (Eq. A3/A5/A6)
$$ φ_{h\,local} = \mathrm{ifelse}\left(\mathrm{>}\left(L, zero_{length}\right), \mathrm{ifelse}\left(\mathrm{<=}\left(ζ, 1\right), 1 + 5 \cdot ζ, 5 + ζ\right), \mathrm{ifelse}\left(\mathrm{<}\left(L, zero_{length}\right), \frac{1}{\sqrt{1 - 15 \cdot ζ}}, 1.0\right)\right) $$Dimensionless temperature gradient at local height z (Eq. A3/A5/A6)
$$ φ_{m\,local} = \mathrm{ifelse}\left(\mathrm{>}\left(L, zero_{length}\right), \mathrm{ifelse}\left(\mathrm{<=}\left(ζ, 1\right), 1 + 5 \cdot ζ, 5 + ζ\right), \mathrm{ifelse}\left(\mathrm{<}\left(L, zero_{length}\right), \frac{1}{\left(1 - 15 \cdot ζ\right)^{\frac{1.0}{3.0}}}, 1.0\right)\right) $$Dimensionless wind shear at local height z (Eq. A2/A4/A8)
$$ w_{m\,outer} = \left(\frac{u_{star}}{w_{ref}}^{3} + c_{1} \cdot \frac{w_{star}}{w_{ref}}^{3}\right)^{\frac{1.0}{3.0}} \cdot w_{ref} $$Outer-layer momentum velocity scale (Eq. A11)
$$ w_{m} = \mathrm{ifelse}\left(\mathrm{<=}\left(η, ε_{sl}\right), \frac{u_{star}}{φ_{m\,local}}, w_{m\,outer}\right) $$Velocity scale for momentum, height-dependent (Eq. A7/A11)
$$ Pr = \frac{φ_{h}}{φ_{m}} + a_{coeff} \cdot κ \cdot ε_{sl} \cdot \frac{w_{star}}{w_{m\,outer}} $$Turbulent Prandtl number (Eq. A13)
$$ w_{t} = \mathrm{ifelse}\left(\mathrm{<=}\left(η, ε_{sl}\right), \frac{u_{star}}{φ_{h\,local}}, \frac{w_{m}}{Pr}\right) $$Velocity scale for scalars, height-dependent (Eq. A1/A10)
$$ Kc = κ \cdot w_{t} \cdot z \cdot \left(1 - η\right)^{2} $$Eddy diffusivity profile (Eq. 3.9)
$$ γc = \mathrm{ifelse}\left(\mathrm{>}\left(wθ_{v0}, zero_{heat\,flux}\right), \frac{a_{coeff} \cdot w_{star} \cdot wC_{0}}{w_{m\,outer}^{2} \cdot h}, γc_{zero}\right) $$Nonlocal countergradient transport term (Eq. 3.10)
Analyses
eddy_diffusivity_profile
Nonlocal-scheme eddy-diffusivity profile Kc(z) through an unstable convective boundary layer of depth h=1000 m (L=-100 m, wθ_v0=0.1 K·m/s). Reproduces the ‘Eddy Diffusivity Profile (Eq. 3.9)’ figure of the AtmosphericDynamics.jl holtslag_boville docs page: Kc = κ·wₜ·z·(1−z/h)² vanishes at the surface and the BL top with a maximum near z ≈ h/3.
Time span: 0.0 → 1.0
Swept over: z
convective_velocity_scale
Convective velocity scale w_star as a function of surface virtual heat flux and boundary-layer depth, w_star = ((g/θ_v0)·wθ_v0·h)^(1/3) (Eq. A12). Reproduces the ‘Convective Velocity Scale Dependence (Eq. A12)’ figure of the AtmosphericDynamics.jl holtslag_boville docs page as a two-dimensional sweep over (wθ_v0, h): w_star grows with both the surface heat flux and the boundary-layer depth.
Time span: 0.0 → 1.0
Swept over: wθ_v0, h
Raw .esm
Raw .esm JSON (this component)
{
"reference": {
"doi": "10.1175/1520-0442(1993)006<1825:LVNBLD>2.0.CO;2",
"citation": "Holtslag & Boville, 1993",
"notes": "source_ref: AtmosphericDynamics.jl@816cc3918e1eef2c2ee6d8fa8dd73c8f13a85a47:src/holtslag_boville_1993.jl:290\nUpstream `HoltslagBovilleNonlocalABL` is an MTK `@component` that returns `System(eqs, t)` with 14 time-indexed `@variables`, each fixed by one algebraic equation (no time derivative appears), and 7 `@parameters` of inputs. It is migrated as an algebraic (observed-only) model: the 7 inputs are `parameter`s, the 14 turbulence quantities are `state` variables defined by the algebraic `equations` — the same serialization pattern as components/atmospheric_dynamics/sp_ch1/ideal_gas.esm and components/wildland_fire/rothermel/wind_limit.esm. No ODE integration occurs; tests assert the closed-form solution at t=0.\nThe 11 upstream `@constants` (g, κ, a_coeff, c_1, ε_sl, w_ref, zero_heat_flux, zero_length, γc_zero, zero_velocity, L_min) are preserved as `parameter` declarations with their upstream defaults, matching the MTK-`@constants`-to-ESM-`parameter` convention used in wind_limit.esm. The `zero_heat_flux`/`zero_length`/`zero_velocity`/`γc_zero` constants are kept rather than inlined as `0.0` literals so the `ifelse` comparison operands and stable-branch values carry units matching the quantities they are compared with (K*m/s, m, m/s, 1/m).\nVariable names: Unicode subscripts in the upstream identifiers are transliterated to ASCII (φₕ→φ_h, φₘ→φ_m, φₕ_local→φ_h_local, φₘ_local→φ_m_local, wₘ→w_m, wₘ_outer→w_m_outer, wₜ→w_t, wθᵥ₀→wθ_v0, wC₀→wC_0, θᵥ₀→θ_v0, c₁→c_1); Greek base letters (φ, ζ, η, κ, ε, γ, θ) are retained, following the convention in components/urban_canopy/urban_radiation.esm.\nRegularization: ζ = z/L and ζ_ε = ε_sl·h/L are guarded for small |L| — when abs(L) ≤ L_min the denominator L is replaced by a value of magnitude L_min carrying L's sign. Upstream encodes this branchlessly as L_min·sign(L) + L_min·(1−abs(sign(L))); it is re-expressed here as the mathematically identical ifelse(L < 0, −L_min, L_min) because the ESS Julia MTK binding (EarthSciASTMTKExt.jl) does not implement the `sign` operator that esm-spec §4.2 lists — the two forms agree exactly for L>0 (+L_min), L<0 (−L_min) and L=0 (+L_min). The regularized branch is never selected by the inline tests (all use |L| ≥ 50 m); it only needs to compile and stay finite.\nTolerance rationale (model-level rel=1e-6): observed-only algebraic system (no ODE integration); every output is a closed-form composition of arithmetic, sqrt, cube-root (^1/3) and ifelse selections in float64. 1e-6 is far above float64 round-off accumulated over the ~10-deep expression chains (w_t depends on w_m and Pr, which depend on the φ profiles and w_star) and below any physical-discrimination scale on the m/s, m^2/s and 1/m output quantities. Exact-zero expectations (w_star and γc in stable conditions, where the `ifelse` selects the zero-velocity / γc_zero constant) carry a per-assertion abs=1e-12 tolerance."
},
"tolerance": {
"rel": 1e-06
},
"variables": {
"z": {
"type": "parameter",
"units": "m",
"default": 100.0,
"description": "Height above surface"
},
"h": {
"type": "parameter",
"units": "m",
"default": 1000.0,
"description": "Boundary layer height"
},
"u_star": {
"type": "parameter",
"units": "m/s",
"default": 0.3,
"description": "Friction velocity"
},
"wθ_v0": {
"type": "parameter",
"units": "K*m/s",
"default": 0.05,
"description": "Surface virtual heat flux (w'θ_v')_0"
},
"wC_0": {
"type": "parameter",
"units": "m/s",
"default": 1e-05,
"description": "Surface kinematic flux of scalar C (e.g. mixing ratio)"
},
"θ_v0": {
"type": "parameter",
"units": "K",
"default": 300.0,
"description": "Virtual potential temperature at surface"
},
"L": {
"type": "parameter",
"units": "m",
"default": -100.0,
"description": "Obukhov length (negative for unstable, positive for stable)"
},
"g": {
"type": "parameter",
"units": "m/s^2",
"default": 9.81,
"description": "Gravitational acceleration"
},
"κ": {
"type": "parameter",
"units": "1",
"default": 0.4,
"description": "von Karman constant"
},
"a_coeff": {
"type": "parameter",
"units": "1",
"default": 7.2,
"description": "Nonlocal transport constant (Eq. A14: a ≈ 7.2)"
},
"c_1": {
"type": "parameter",
"units": "1",
"default": 0.6,
"description": "Velocity scale constant (Eq. A11)"
},
"ε_sl": {
"type": "parameter",
"units": "1",
"default": 0.1,
"description": "Surface layer fraction of boundary layer height"
},
"w_ref": {
"type": "parameter",
"units": "m/s",
"default": 1.0,
"description": "Reference velocity scale for non-dimensionalizing the cube-root arguments"
},
"zero_heat_flux": {
"type": "parameter",
"units": "K*m/s",
"default": 0.0,
"description": "Zero heat flux, comparison constant for the unstable/stable branch"
},
"zero_length": {
"type": "parameter",
"units": "m",
"default": 0.0,
"description": "Zero length, comparison constant for the L sign branch"
},
"γc_zero": {
"type": "parameter",
"units": "1/m",
"default": 0.0,
"description": "Zero nonlocal transport term (stable-branch value of γc)"
},
"zero_velocity": {
"type": "parameter",
"units": "m/s",
"default": 0.0,
"description": "Zero velocity (stable-branch value of w_star)"
},
"L_min": {
"type": "parameter",
"units": "m",
"default": 1.0,
"description": "Minimum absolute Obukhov length for regularization of z/L"
},
"ζ": {
"type": "unknown",
"units": "1",
"description": "Stability parameter z/L at local height"
},
"η": {
"type": "unknown",
"units": "1",
"description": "Relative height z/h"
},
"ζ_ε": {
"type": "unknown",
"units": "1",
"description": "Stability parameter ε_sl·h/L at the surface layer top"
},
"w_star": {
"type": "unknown",
"units": "m/s",
"description": "Convective velocity scale (Eq. A12)"
},
"φ_m": {
"type": "unknown",
"units": "1",
"description": "Dimensionless wind shear at the surface layer top (Eq. A2/A4/A8)"
},
"φ_h": {
"type": "unknown",
"units": "1",
"description": "Dimensionless temperature gradient at the surface layer top (Eq. A3/A5/A6)"
},
"φ_h_local": {
"type": "unknown",
"units": "1",
"description": "Dimensionless temperature gradient at local height z (Eq. A3/A5/A6)"
},
"φ_m_local": {
"type": "unknown",
"units": "1",
"description": "Dimensionless wind shear at local height z (Eq. A2/A4/A8)"
},
"w_m_outer": {
"type": "unknown",
"units": "m/s",
"description": "Outer-layer momentum velocity scale (Eq. A11)"
},
"w_m": {
"type": "unknown",
"units": "m/s",
"description": "Velocity scale for momentum, height-dependent (Eq. A7/A11)"
},
"Pr": {
"type": "unknown",
"units": "1",
"description": "Turbulent Prandtl number (Eq. A13)"
},
"w_t": {
"type": "unknown",
"units": "m/s",
"description": "Velocity scale for scalars, height-dependent (Eq. A1/A10)"
},
"Kc": {
"type": "unknown",
"units": "m^2/s",
"description": "Eddy diffusivity profile (Eq. 3.9)"
},
"γc": {
"type": "unknown",
"units": "1/m",
"description": "Nonlocal countergradient transport term (Eq. 3.10)"
}
},
"equations": [
{
"lhs": "ζ",
"rhs": {
"op": "ifelse",
"args": [
{
"op": ">",
"args": [
{
"op": "abs",
"args": [
"L"
]
},
"L_min"
]
},
{
"op": "/",
"args": [
"z",
"L"
]
},
{
"op": "/",
"args": [
"z",
{
"op": "ifelse",
"args": [
{
"op": "<",
"args": [
"L",
"zero_length"
]
},
{
"op": "-",
"args": [
"L_min"
]
},
"L_min"
]
}
]
}
]
}
},
{
"lhs": "η",
"rhs": {
"op": "/",
"args": [
"z",
"h"
]
}
},
{
"lhs": "ζ_ε",
"rhs": {
"op": "ifelse",
"args": [
{
"op": ">",
"args": [
{
"op": "abs",
"args": [
"L"
]
},
"L_min"
]
},
{
"op": "/",
"args": [
{
"op": "*",
"args": [
"ε_sl",
"h"
]
},
"L"
]
},
{
"op": "/",
"args": [
{
"op": "*",
"args": [
"ε_sl",
"h"
]
},
{
"op": "ifelse",
"args": [
{
"op": "<",
"args": [
"L",
"zero_length"
]
},
{
"op": "-",
"args": [
"L_min"
]
},
"L_min"
]
}
]
}
]
}
},
{
"lhs": "w_star",
"rhs": {
"op": "ifelse",
"args": [
{
"op": ">",
"args": [
"wθ_v0",
"zero_heat_flux"
]
},
{
"op": "*",
"args": [
{
"op": "^",
"args": [
{
"op": "/",
"args": [
{
"op": "*",
"args": [
{
"op": "/",
"args": [
"g",
"θ_v0"
]
},
"wθ_v0",
"h"
]
},
{
"op": "^",
"args": [
"w_ref",
3
]
}
]
},
{
"op": "/",
"args": [
1.0,
3.0
]
}
]
},
"w_ref"
]
},
"zero_velocity"
]
}
},
{
"lhs": "φ_m",
"rhs": {
"op": "ifelse",
"args": [
{
"op": ">",
"args": [
"L",
"zero_length"
]
},
{
"op": "ifelse",
"args": [
{
"op": "<=",
"args": [
"ζ_ε",
1
]
},
{
"op": "+",
"args": [
1,
{
"op": "*",
"args": [
5,
"ζ_ε"
]
}
]
},
{
"op": "+",
"args": [
5,
"ζ_ε"
]
}
]
},
{
"op": "ifelse",
"args": [
{
"op": "<",
"args": [
"L",
"zero_length"
]
},
{
"op": "/",
"args": [
1,
{
"op": "^",
"args": [
{
"op": "-",
"args": [
1,
{
"op": "*",
"args": [
15,
"ζ_ε"
]
}
]
},
{
"op": "/",
"args": [
1.0,
3.0
]
}
]
}
]
},
1.0
]
}
]
}
},
{
"lhs": "φ_h",
"rhs": {
"op": "ifelse",
"args": [
{
"op": ">",
"args": [
"L",
"zero_length"
]
},
{
"op": "ifelse",
"args": [
{
"op": "<=",
"args": [
"ζ_ε",
1
]
},
{
"op": "+",
"args": [
1,
{
"op": "*",
"args": [
5,
"ζ_ε"
]
}
]
},
{
"op": "+",
"args": [
5,
"ζ_ε"
]
}
]
},
{
"op": "ifelse",
"args": [
{
"op": "<",
"args": [
"L",
"zero_length"
]
},
{
"op": "/",
"args": [
1,
{
"op": "sqrt",
"args": [
{
"op": "-",
"args": [
1,
{
"op": "*",
"args": [
15,
"ζ_ε"
]
}
]
}
]
}
]
},
1.0
]
}
]
}
},
{
"lhs": "φ_h_local",
"rhs": {
"op": "ifelse",
"args": [
{
"op": ">",
"args": [
"L",
"zero_length"
]
},
{
"op": "ifelse",
"args": [
{
"op": "<=",
"args": [
"ζ",
1
]
},
{
"op": "+",
"args": [
1,
{
"op": "*",
"args": [
5,
"ζ"
]
}
]
},
{
"op": "+",
"args": [
5,
"ζ"
]
}
]
},
{
"op": "ifelse",
"args": [
{
"op": "<",
"args": [
"L",
"zero_length"
]
},
{
"op": "/",
"args": [
1,
{
"op": "sqrt",
"args": [
{
"op": "-",
"args": [
1,
{
"op": "*",
"args": [
15,
"ζ"
]
}
]
}
]
}
]
},
1.0
]
}
]
}
},
{
"lhs": "φ_m_local",
"rhs": {
"op": "ifelse",
"args": [
{
"op": ">",
"args": [
"L",
"zero_length"
]
},
{
"op": "ifelse",
"args": [
{
"op": "<=",
"args": [
"ζ",
1
]
},
{
"op": "+",
"args": [
1,
{
"op": "*",
"args": [
5,
"ζ"
]
}
]
},
{
"op": "+",
"args": [
5,
"ζ"
]
}
]
},
{
"op": "ifelse",
"args": [
{
"op": "<",
"args": [
"L",
"zero_length"
]
},
{
"op": "/",
"args": [
1,
{
"op": "^",
"args": [
{
"op": "-",
"args": [
1,
{
"op": "*",
"args": [
15,
"ζ"
]
}
]
},
{
"op": "/",
"args": [
1.0,
3.0
]
}
]
}
]
},
1.0
]
}
]
}
},
{
"lhs": "w_m_outer",
"rhs": {
"op": "*",
"args": [
{
"op": "^",
"args": [
{
"op": "+",
"args": [
{
"op": "^",
"args": [
{
"op": "/",
"args": [
"u_star",
"w_ref"
]
},
3
]
},
{
"op": "*",
"args": [
"c_1",
{
"op": "^",
"args": [
{
"op": "/",
"args": [
"w_star",
"w_ref"
]
},
3
]
}
]
}
]
},
{
"op": "/",
"args": [
1.0,
3.0
]
}
]
},
"w_ref"
]
}
},
{
"lhs": "w_m",
"rhs": {
"op": "ifelse",
"args": [
{
"op": "<=",
"args": [
"η",
"ε_sl"
]
},
{
"op": "/",
"args": [
"u_star",
"φ_m_local"
]
},
"w_m_outer"
]
}
},
{
"lhs": "Pr",
"rhs": {
"op": "+",
"args": [
{
"op": "/",
"args": [
"φ_h",
"φ_m"
]
},
{
"op": "*",
"args": [
"a_coeff",
"κ",
"ε_sl",
{
"op": "/",
"args": [
"w_star",
"w_m_outer"
]
}
]
}
]
}
},
{
"lhs": "w_t",
"rhs": {
"op": "ifelse",
"args": [
{
"op": "<=",
"args": [
"η",
"ε_sl"
]
},
{
"op": "/",
"args": [
"u_star",
"φ_h_local"
]
},
{
"op": "/",
"args": [
"w_m",
"Pr"
]
}
]
}
},
{
"lhs": "Kc",
"rhs": {
"op": "*",
"args": [
"κ",
"w_t",
"z",
{
"op": "^",
"args": [
{
"op": "-",
"args": [
1,
"η"
]
},
2
]
}
]
}
},
{
"lhs": "γc",
"rhs": {
"op": "ifelse",
"args": [
{
"op": ">",
"args": [
"wθ_v0",
"zero_heat_flux"
]
},
{
"op": "/",
"args": [
{
"op": "*",
"args": [
"a_coeff",
"w_star",
"wC_0"
]
},
{
"op": "*",
"args": [
{
"op": "^",
"args": [
"w_m_outer",
2
]
},
"h"
]
}
]
},
"γc_zero"
]
}
}
],
"tests": [
{
"id": "default_parameters",
"description": "All seven inputs at the upstream component defaults (z=100 m, h=1000 m, u_star=0.3 m/s, wθ_v0=0.05 K·m/s, wC_0=1e-5 m/s, θ_v0=300 K, L=-100 m). z/h = 0.1 lands exactly on the surface-layer/outer-layer boundary η = ε_sl, exercising the `η ≤ ε_sl` branch of w_m and w_t.",
"parameter_overrides": {
"z": 100.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": 0.05,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": -100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": -1.0
},
{
"variable": "η",
"time": 0.0,
"expected": 0.1
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": -1.0
},
{
"variable": "w_star",
"time": 0.0,
"expected": 1.1780740391008697
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 0.39685026299204984
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 0.25
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 0.25
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 0.39685026299204984
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 1.002659586992917
},
{
"variable": "w_m",
"time": 0.0,
"expected": 0.7559526299369239
},
{
"variable": "Pr",
"time": 0.0,
"expected": 0.9683458829118514
},
{
"variable": "w_t",
"time": 0.0,
"expected": 1.2
},
{
"variable": "Kc",
"time": 0.0,
"expected": 38.88
},
{
"variable": "γc",
"time": 0.0,
"expected": 8.437194496371112e-08
}
]
},
{
"id": "unstable_outer_z500_Lm100",
"description": "Canonical unstable convective boundary layer sampled in the outer layer (z=500 m, η=0.5 > ε_sl, L=-100 m). Reproduces the upstream `convective velocity scale Eq. A12`, `momentum velocity scale Eq. A11`, `Prandtl number Eq. A13` and `outer layer wₜ Eq. A10` test points: w_star ≈ 1.48 m/s, and w_m equals w_m_outer in the outer layer.",
"parameter_overrides": {
"z": 500.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": 0.1,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": -100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": -5.0
},
{
"variable": "η",
"time": 0.0,
"expected": 0.5
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": -1.0
},
{
"variable": "w_star",
"time": 0.0,
"expected": 1.4842802801978616
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 0.39685026299204984
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 0.25
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 0.11470786693528087
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 0.2360815978543417
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "w_m",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "Pr",
"time": 0.0,
"expected": 0.9698701618539511
},
{
"variable": "w_t",
"time": 0.0,
"expected": 1.2966755720281316
},
{
"variable": "Kc",
"time": 0.0,
"expected": 64.83377860140659
},
{
"variable": "γc",
"time": 0.0,
"expected": 6.757072186590978e-08
}
]
},
{
"id": "kc_profile_surface_z50",
"description": "Eddy-diffusivity profile, near-surface sample (z=50 m, η=0.05, surface layer) under unstable conditions. Upstream `eddy diffusivity profile Eq. 3.9` low point — Kc is small near the surface.",
"parameter_overrides": {
"z": 50.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": 0.1,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": -100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": -0.5
},
{
"variable": "η",
"time": 0.0,
"expected": 0.05
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": -1.0
},
{
"variable": "w_star",
"time": 0.0,
"expected": 1.4842802801978616
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 0.39685026299204984
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 0.25
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 0.34299717028501764
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 0.4899973050296446
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "w_m",
"time": 0.0,
"expected": 0.6122482652876022
},
{
"variable": "Pr",
"time": 0.0,
"expected": 0.9698701618539511
},
{
"variable": "w_t",
"time": 0.0,
"expected": 0.8746427842267951
},
{
"variable": "Kc",
"time": 0.0,
"expected": 15.78730225529365
},
{
"variable": "γc",
"time": 0.0,
"expected": 6.757072186590978e-08
}
]
},
{
"id": "kc_profile_mid_z333",
"description": "Eddy-diffusivity profile, mid-boundary-layer sample (z=333 m ≈ h/3, outer layer) where the Kc = κ·wₜ·z·(1−z/h)² profile peaks. Upstream `eddy diffusivity profile Eq. 3.9` mid point.",
"parameter_overrides": {
"z": 333.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": 0.1,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": -100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": -3.33
},
{
"variable": "η",
"time": 0.0,
"expected": 0.333
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": -1.0
},
{
"variable": "w_star",
"time": 0.0,
"expected": 1.4842802801978616
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 0.39685026299204984
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 0.25
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 0.14009670009611716
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 0.26974408938034933
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "w_m",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "Pr",
"time": 0.0,
"expected": 0.9698701618539511
},
{
"variable": "w_t",
"time": 0.0,
"expected": 1.2966755720281316
},
{
"variable": "Kc",
"time": 0.0,
"expected": 76.83997624872794
},
{
"variable": "γc",
"time": 0.0,
"expected": 6.757072186590978e-08
}
]
},
{
"id": "kc_profile_top_z900",
"description": "Eddy-diffusivity profile, near-BL-top sample (z=900 m, η=0.9). The (1−η)² factor drives Kc back toward zero approaching the boundary-layer top. Upstream `eddy diffusivity profile Eq. 3.9` high point.",
"parameter_overrides": {
"z": 900.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": 0.1,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": -100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": -9.0
},
{
"variable": "η",
"time": 0.0,
"expected": 0.9
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": -1.0
},
{
"variable": "w_star",
"time": 0.0,
"expected": 1.4842802801978616
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 0.39685026299204984
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 0.25
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 0.08574929257125441
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 0.19445555936641015
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "w_m",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "Pr",
"time": 0.0,
"expected": 0.9698701618539511
},
{
"variable": "w_t",
"time": 0.0,
"expected": 1.2966755720281316
},
{
"variable": "Kc",
"time": 0.0,
"expected": 4.668032059301272
},
{
"variable": "γc",
"time": 0.0,
"expected": 6.757072186590978e-08
}
]
},
{
"id": "kc_verify_z400",
"description": "Numerical verification of the eddy-diffusivity profile Kc = κ·wₜ·z·(1−z/h)² at z=400 m, h=1000 m. Upstream `Kc numerical verification Eq. 3.9`.",
"parameter_overrides": {
"z": 400.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": 0.1,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": -100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": -4.0
},
{
"variable": "η",
"time": 0.0,
"expected": 0.4
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": -1.0
},
{
"variable": "w_star",
"time": 0.0,
"expected": 1.4842802801978616
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 0.39685026299204984
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 0.25
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 0.12803687993289598
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 0.2540329520093663
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "w_m",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "Pr",
"time": 0.0,
"expected": 0.9698701618539511
},
{
"variable": "w_t",
"time": 0.0,
"expected": 1.2966755720281316
},
{
"variable": "Kc",
"time": 0.0,
"expected": 74.68851294882037
},
{
"variable": "γc",
"time": 0.0,
"expected": 6.757072186590978e-08
}
]
},
{
"id": "near_neutral_large_L",
"description": "Near-neutral limit: very large |L| (L=-10000 m), tiny surface heat flux (wθ_v0=0.001 K·m/s) and strong wind (u_star=1 m/s). The convective scale w_star is small and the turbulent Prandtl number Pr ≈ 1. Upstream `Prandtl number Eq. A13` near-neutral point.",
"parameter_overrides": {
"z": 500.0,
"h": 1000.0,
"u_star": 1.0,
"wθ_v0": 0.001,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": -10000.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": -0.05
},
{
"variable": "η",
"time": 0.0,
"expected": 0.5
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": -0.01
},
{
"variable": "w_star",
"time": 0.0,
"expected": 0.31977849253885177
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 0.9544812172391357
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 0.9325048082403138
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 0.7559289460184544
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 0.8298265333662433
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 1.006497688598828
},
{
"variable": "w_m",
"time": 0.0,
"expected": 1.006497688598828
},
{
"variable": "Pr",
"time": 0.0,
"expected": 1.0684772024561595
},
{
"variable": "w_t",
"time": 0.0,
"expected": 0.9419926660907165
},
{
"variable": "Kc",
"time": 0.0,
"expected": 47.09963330453583
},
{
"variable": "γc",
"time": 0.0,
"expected": 2.2727736391485038e-08
}
]
},
{
"id": "gamma_c_unstable",
"description": "Nonlocal countergradient transport in unstable conditions with an order-of-magnitude larger scalar surface flux (wC_0=1e-4 m/s): γc > 0. Upstream `nonlocal transport term Eq. 3.10` unstable point.",
"parameter_overrides": {
"z": 500.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": 0.1,
"wC_0": 0.0001,
"θ_v0": 300.0,
"L": -100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": -5.0
},
{
"variable": "η",
"time": 0.0,
"expected": 0.5
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": -1.0
},
{
"variable": "w_star",
"time": 0.0,
"expected": 1.4842802801978616
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 0.39685026299204984
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 0.25
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 0.11470786693528087
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 0.2360815978543417
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "w_m",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "Pr",
"time": 0.0,
"expected": 0.9698701618539511
},
{
"variable": "w_t",
"time": 0.0,
"expected": 1.2966755720281316
},
{
"variable": "Kc",
"time": 0.0,
"expected": 64.83377860140659
},
{
"variable": "γc",
"time": 0.0,
"expected": 6.757072186590977e-07
}
]
},
{
"id": "gamma_c_stable",
"description": "Stable conditions (L=+100 m, wθ_v0=-0.05 K·m/s): the convective scale w_star and the nonlocal term γc both collapse to zero — the nonlocal scheme adds no countergradient transport. Upstream `nonlocal transport term Eq. 3.10` stable point.",
"parameter_overrides": {
"z": 500.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": -0.05,
"wC_0": 0.0001,
"θ_v0": 300.0,
"L": 100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": 5.0
},
{
"variable": "η",
"time": 0.0,
"expected": 0.5
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": 1.0
},
{
"variable": "w_star",
"time": 0.0,
"expected": 0.0,
"tolerance": {
"abs": 1e-12
}
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 6.0
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 6.0
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 10.0
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 10.0
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 0.3
},
{
"variable": "w_m",
"time": 0.0,
"expected": 0.3
},
{
"variable": "Pr",
"time": 0.0,
"expected": 1.0
},
{
"variable": "w_t",
"time": 0.0,
"expected": 0.3
},
{
"variable": "Kc",
"time": 0.0,
"expected": 15.0
},
{
"variable": "γc",
"time": 0.0,
"expected": 0.0,
"tolerance": {
"abs": 1e-12
}
}
]
},
{
"id": "phi_stable_L100_z50",
"description": "Stable surface layer, L=+100 m ⇒ ζ_ε = ε_sl·h/L = 1.0, so φ_m = φ_h = 1 + 5·1 = 6 (Eq. A2/A3 stable branch). Upstream `phi_h and phi_m Eq. A2-A6` stable case.",
"parameter_overrides": {
"z": 50.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": -0.05,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": 100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": 0.5
},
{
"variable": "η",
"time": 0.0,
"expected": 0.05
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": 1.0
},
{
"variable": "w_star",
"time": 0.0,
"expected": 0.0,
"tolerance": {
"abs": 1e-12
}
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 6.0
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 6.0
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 3.5
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 3.5
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 0.3
},
{
"variable": "w_m",
"time": 0.0,
"expected": 0.08571428571428572
},
{
"variable": "Pr",
"time": 0.0,
"expected": 1.0
},
{
"variable": "w_t",
"time": 0.0,
"expected": 0.08571428571428572
},
{
"variable": "Kc",
"time": 0.0,
"expected": 1.5471428571428572
},
{
"variable": "γc",
"time": 0.0,
"expected": 0.0,
"tolerance": {
"abs": 1e-12
}
}
]
},
{
"id": "phi_very_stable_L50_z50",
"description": "Very stable surface layer, L=+50 m ⇒ ζ_ε = 2.0 > 1, so φ_m = φ_h = 5 + 2 = 7 (Eq. A4/A5 very-stable branch, ζ_ε > 1). Upstream `phi_h and phi_m Eq. A2-A6` very-stable case.",
"parameter_overrides": {
"z": 50.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": -0.05,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": 50.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": 1.0
},
{
"variable": "η",
"time": 0.0,
"expected": 0.05
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": 2.0
},
{
"variable": "w_star",
"time": 0.0,
"expected": 0.0,
"tolerance": {
"abs": 1e-12
}
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 7.0
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 7.0
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 6.0
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 6.0
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 0.3
},
{
"variable": "w_m",
"time": 0.0,
"expected": 0.049999999999999996
},
{
"variable": "Pr",
"time": 0.0,
"expected": 1.0
},
{
"variable": "w_t",
"time": 0.0,
"expected": 0.049999999999999996
},
{
"variable": "Kc",
"time": 0.0,
"expected": 0.9025
},
{
"variable": "γc",
"time": 0.0,
"expected": 0.0,
"tolerance": {
"abs": 1e-12
}
}
]
},
{
"id": "phi_unstable_Lm100_z50",
"description": "Unstable surface layer, L=-100 m ⇒ ζ_ε = -1.0, so φ_h = (1−15·ζ_ε)^(−1/2) = 16^(−1/2) = 0.25 (Eq. A6) and φ_m = 16^(−1/3) (Eq. A8). Upstream `phi_h and phi_m Eq. A2-A6` unstable case.",
"parameter_overrides": {
"z": 50.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": 0.1,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": -100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": -0.5
},
{
"variable": "η",
"time": 0.0,
"expected": 0.05
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": -1.0
},
{
"variable": "w_star",
"time": 0.0,
"expected": 1.4842802801978616
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 0.39685026299204984
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 0.25
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 0.34299717028501764
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 0.4899973050296446
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "w_m",
"time": 0.0,
"expected": 0.6122482652876022
},
{
"variable": "Pr",
"time": 0.0,
"expected": 0.9698701618539511
},
{
"variable": "w_t",
"time": 0.0,
"expected": 0.8746427842267951
},
{
"variable": "Kc",
"time": 0.0,
"expected": 15.78730225529365
},
{
"variable": "γc",
"time": 0.0,
"expected": 6.757072186590978e-08
}
]
},
{
"id": "surface_layer_Lm200_z50",
"description": "Surface layer (z=50 m, η=0.05 ≤ ε_sl) with L=-200 m. Exercises the local-height profiles φ_h_local, φ_m_local evaluated at ζ = z/L and the surface-layer velocity scales w_m = u_star/φ_m_local (Eq. A7) and w_t = u_star/φ_h_local (Eq. A1). Upstream `phi_h_local and phi_m_local Eq. A1/A7`, `surface layer w_m Eq. A7`, `surface layer wₜ Eq. A1`.",
"parameter_overrides": {
"z": 50.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": 0.1,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": -200.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": -0.25
},
{
"variable": "η",
"time": 0.0,
"expected": 0.05
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": -0.5
},
{
"variable": "w_star",
"time": 0.0,
"expected": 1.4842802801978616
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 0.4899973050296446
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 0.34299717028501764
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 0.4588314677411235
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 0.5948883492590029
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 1.2576069469149886
},
{
"variable": "w_m",
"time": 0.0,
"expected": 0.5042963110198445
},
{
"variable": "Pr",
"time": 0.0,
"expected": 1.0399077119250424
},
{
"variable": "w_t",
"time": 0.0,
"expected": 0.653834841531101
},
{
"variable": "Kc",
"time": 0.0,
"expected": 11.801718889636374
},
{
"variable": "γc",
"time": 0.0,
"expected": 6.757072186590978e-08
}
]
},
{
"id": "stable_outer_z500_L100",
"description": "Stable outer layer (z=500 m, η=0.5, L=+100 m): w_star=0, γc=0, and the outer-layer momentum scale collapses to w_m_outer = (u_star³)^(1/3) = u_star = 0.3 m/s, with w_m = w_m_outer. Upstream `stable conditions` test.",
"parameter_overrides": {
"z": 500.0,
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": -0.05,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": 100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ζ",
"time": 0.0,
"expected": 5.0
},
{
"variable": "η",
"time": 0.0,
"expected": 0.5
},
{
"variable": "ζ_ε",
"time": 0.0,
"expected": 1.0
},
{
"variable": "w_star",
"time": 0.0,
"expected": 0.0,
"tolerance": {
"abs": 1e-12
}
},
{
"variable": "φ_m",
"time": 0.0,
"expected": 6.0
},
{
"variable": "φ_h",
"time": 0.0,
"expected": 6.0
},
{
"variable": "φ_h_local",
"time": 0.0,
"expected": 10.0
},
{
"variable": "φ_m_local",
"time": 0.0,
"expected": 10.0
},
{
"variable": "w_m_outer",
"time": 0.0,
"expected": 0.3
},
{
"variable": "w_m",
"time": 0.0,
"expected": 0.3
},
{
"variable": "Pr",
"time": 0.0,
"expected": 1.0
},
{
"variable": "w_t",
"time": 0.0,
"expected": 0.3
},
{
"variable": "Kc",
"time": 0.0,
"expected": 15.0
},
{
"variable": "γc",
"time": 0.0,
"expected": 0.0,
"tolerance": {
"abs": 1e-12
}
}
]
}
],
"analyses": [
{
"id": "eddy_diffusivity_profile",
"description": "Nonlocal-scheme eddy-diffusivity profile Kc(z) through an unstable convective boundary layer of depth h=1000 m (L=-100 m, wθ_v0=0.1 K·m/s). Reproduces the 'Eddy Diffusivity Profile (Eq. 3.9)' figure of the AtmosphericDynamics.jl holtslag_boville docs page: Kc = κ·wₜ·z·(1−z/h)² vanishes at the surface and the BL top with a maximum near z ≈ h/3.",
"initial_state": {},
"parameters": {
"h": 1000.0,
"u_star": 0.3,
"wθ_v0": 0.1,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": -100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"parameter_sweep": {
"type": "cartesian",
"dimensions": [
{
"parameter": "z",
"range": {
"start": 50.0,
"stop": 950.0,
"count": 19,
"scale": "linear"
}
}
]
},
"plots": [
{
"id": "kc_vs_height",
"type": "line",
"description": "Eddy diffusivity Kc as a function of height z (h=1000 m, unstable).",
"x": {
"variable": "z",
"label": "Height z (m)"
},
"y": {
"variable": "Kc",
"label": "Eddy diffusivity Kc (m^2/s)"
}
}
]
},
{
"id": "convective_velocity_scale",
"description": "Convective velocity scale w_star as a function of surface virtual heat flux and boundary-layer depth, w_star = ((g/θ_v0)·wθ_v0·h)^(1/3) (Eq. A12). Reproduces the 'Convective Velocity Scale Dependence (Eq. A12)' figure of the AtmosphericDynamics.jl holtslag_boville docs page as a two-dimensional sweep over (wθ_v0, h): w_star grows with both the surface heat flux and the boundary-layer depth.",
"initial_state": {},
"parameters": {
"z": 500.0,
"u_star": 0.3,
"wC_0": 1e-05,
"θ_v0": 300.0,
"L": -100.0
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"parameter_sweep": {
"type": "cartesian",
"dimensions": [
{
"parameter": "wθ_v0",
"range": {
"start": 0.01,
"stop": 0.2,
"count": 20,
"scale": "linear"
}
},
{
"parameter": "h",
"range": {
"start": 500.0,
"stop": 2000.0,
"count": 16,
"scale": "linear"
}
}
]
},
"plots": [
{
"id": "w_star_heatmap",
"type": "heatmap",
"description": "Convective velocity scale w_star over surface heat flux wθ_v0 and BL depth h.",
"x": {
"variable": "wθ_v0",
"label": "Surface virtual heat flux wθ_v0 (K*m/s)"
},
"y": {
"variable": "h",
"label": "Boundary layer height h (m)"
},
"value": {
"variable": "w_star",
"reduce": "final"
}
}
]
}
]
}