Non-equilibrium bulk microphysics scheme, which includes:

• condensation and evaporation of cloud liquid water and deposition and sublimation of cloud ice (relaxation to equilibrium)

τ_relax(liquid)
τ_relax(ice)

• liquid or ice - a type for cloud liquid water or ice

Returns the relaxation timescale for condensation and evaporation of cloud liquid water or the relaxation timescale for sublimation and deposition of cloud ice.

conv_q_vap_to_q_lcl_icl(params, q_sat, q_lcl)
conv_q_vap_to_q_lcl_icl(params, q_sat, q_icl)

• params - a struct with cloud water or ice free parameters
• q_sat - liquid water or ice equilibrium specific contents
• q_lcl, q_icl - current cloud liquid water or cloud ice specific contants

Returns the cloud liquid water tendency due to condensation and evaporation or cloud ice tendency due to sublimation and vapor deposition. The tendency is obtained assuming a relaxation to equilibrium with a constant timescale and is based on the difference between specific contents in equilibrium at the current temperature and the current cloud condensate.

conv_q_vap_to_q_lcl_icl_MM2015(params, tps, q_tot, q_lcl, q_icl, q_rai, q_sno, ρ, T)

• params - a struct with cloud liquid water or ice free parameters
• tps - thermodynamics parameters struct
• q_tot, q_lcl, q_icl, q_rai, q_sno - specific contents of total water, cloud liquid water and ice, rain and snow,
• ρ - air density [kg/m3]
• T - air temperature [K]

Returns the cloud liquid water tendency due to condensation and evaporation or cloud ice tendency due to sublimation and vapor deposition. The formulation is based on Morrison and Grabowski 2008 and Morrison and Milbrandt 2015.

It does NOT screen for small or negative values for humidities, so we suggest applying a limiter of choice to this function, when running it in a model.

terminal_velocity(sediment, vel, ρ, q)

• sediment - a struct with sedimentation type (cloud liquid or ice)
• vel - a struct with terminal velocity parameters
• ρₐ - air density
• q - cloud liquid water or ice specific content

Returns the mass weighted average terminal velocity assuming a monodisperse size distribution with prescribed number concentration. The fall velocity of individual particles is parameterized following Chen et. al 2022, DOI: 10.1016/j.atmosres.2022.106171

Microphysics0M

Zero-moment bulk microphysics scheme that instantly removes moisture above certain threshold. This is equivalent to instanteneous conversion of cloud condensate into precipitation and precipitation fallout with infinite terminal velocity.

remove_precipitation(params_0M, q_lcl, q_icl; q_vap_sat)
remove_precipitation(params_0M, q; q_vap_sat)

• params_0M - a struct with 0-moment parameters
• q - Thermodynamics PhasePartition struct (deprecated)
• q_lcl and q_icl cloud liquid water and cloud ice specific contents
• q_vap_sat - specific humidity at saturation

Returns the tota water (q_tot) tendency due to the removal of precipitation. The tendency is obtained assuming a relaxation with a constant timescale to a state with precipitable water removed. The threshold for when to remove q_tot is defined either by the condensate specific content or supersaturation. The thresholds and the relaxation timescale are defined Parameters0M struct.

One-moment bulk microphysics scheme, which includes:

• terminal velocity of precipitation
• autoconversion of cloud liquid water into rain and of cloud ice into snow
• accretion due to collisions between categories of condensed species
• evaporation and sublimation of hydrometeors
• melting of snow into rain

get_v0(v, ρ)

• v - a struct with bulk 1-moment terminal velocity parameters
• ρ - air density (only for Rain)

Returns the proportionality coefficient in terminal velocity(r/r0).

get_n0(pdf, q_sno, ρ)

• pdf - a struct with parameters for snow, ice, and rain size distribution
• q_sno - snow specific content (only for Snow)
• ρ - air density (only for Snow)

Returns the intercept parameter of the assumed Marshall-Palmer distribution

lambda_inverse(pdf, mass, q, ρ)

• pdf, mass - structs with particle size distribution and mass parameters
• q - specific content of rain, cloud ice or snow
• ρ - air density

Returns the inverse of rate parameter of the assumed size distribution of particles (rain drops, ice crystals, snow crystals). The value is clipped at 1e-8.

terminal_velocity(precip, vel, ρ, q)

• precip - a struct with precipitation type (rain or snow)
• vel - a struct with terminal velocity parameters
• ρ - air density
• q - rain or snow specific content

Returns the mass weighted average terminal velocity assuming a Marshall-Palmer (1948) distribution of particles. Fall velocity of individual rain drops is parameterized:

• assuming an empirical power-law relations for velocity == Blk1MVelType
• following Chen et. al 2022, DOI: 10.1016/j.atmosres.2022.106171, for velocity == Chen2022VelType

conv_q_lcl_to_q_rai(acnv, q_lcl, smooth_transition)

• acnv - 1M autoconversion parameters
• q_lcl - cloud liquid water specific content
• smooth_transition - a flag to switch on smoothing

Returns the q_rai tendency due to collisions between cloud droplets (autoconversion), parametrized following Kessler (1995).

conv_q_icl_to_q_sno_no_supersat(acnv, q_icl, smooth_transition)

• acnv - 1M autoconversion parameters
• q_icl - cloud ice specific content
• smooth_transition - a flag to switch on smoothing

Returns the q_sno tendency due to autoconversion from cloud ice. This is a simplified version of a snow autoconversion rate that can be used in simulations where there is no supersaturation (for example in TC.jl when using saturation adjustment).

conv_q_icl_to_q_sno(ice, aps, tps, q_tot, q_lcl, q_icl, q_rai, q_sno, ρ, T)

• ice - a struct with ice parameters
• aps - a struct with air properties
• tps - a struct with thermodynamics parameters
• q_tot - total water specific content
• q_lcl - cloud liquid water specific content
• q_icl - cloud ice specific content
• q_rai - rain specific content
• q_sno - snow specific content
• ρ - air density
• T - air temperature

Returns the q_sno tendency due to autoconversion from ice. Parameterized following Harrington et al. (1996) and Kaul et al. (2015).

accretion(cloud, precip, vel, ce, q_clo, q_pre, ρ)

• cloud - type for cloud water or cloud ice
• precip - type for rain or snow
• vel - a struct with terminal velocity parameters
• ce - collision efficiency parameters
• q_clo - cloud liquid water or cloud ice specific content
• q_pre - rain water or snow specific content
• ρ - air density

Returns the source of precipitating water (rain or snow) due to collisions with cloud water (liquid or ice).

accretion_rain_sink(rain, ice, vel, ce, q_icl, q_rai, ρ)

• rain - rain type parameters
• ice - ice type parameters
• vel - terminal velocity parameters for rain
• ce - collision efficiency parameters
• q_icl - cloud ice specific content
• q_rai - rain water specific content
• ρ - air density

Returns the sink of rain water (partial source of snow) due to collisions with cloud ice.

accretion_snow_rain(ce, type_i, type_j, blk1m_type_i, blk1m_type_j, q_i, q_j, ρ)

• ce - collision efficiency parameters
• i - snow for temperatures below freezing or rain for temperatures above freezing
• j - rain for temperatures below freezing or snow for temperatures above freezing
• type_i, type_j - a type for snow or rain
• blk1mveltype_ti, blk1mveltype_tj - 1M terminal velocity parameters
• q_ - specific content of snow or rain
• ρ - air density

Returns the accretion rate between rain and snow. Collisions between rain and snow result in snow at temperatures below freezing and in rain at temperatures above freezing.

evaporation_sublimation(params, vel, aps, tps, q_tot, q_lcl, q_icl, q_rai, q_snow, ρ, T)

• params - a struct with rain or snow parameters
• vel - a struct with terminal velocity parameters
• aps - a struct with air parameters
• tps - a struct with thermodynamics parameters
• q_tot - total water specific content
• q_lcl - cloud liquid water specific content
• q_icl - cloud ice specific content
• q_rai - rain specific content
• q_sno - snow specific content
• ρ - air density
• T - air temperature

Returns the tendency due to rain evaporation or snow sublimation.

snow_melt(snow, vel, aps, tps, q_sno, ρ, T)

• snow - snow parameters
• vel - terminal velocity parameters
• aps - air properties
• tps - thermodynamics parameters
• q_sno - snow water specific content
• ρ - air density
• T - air temperature

Returns the tendency due to snow melt.

Double-moment bulk microphysics parametrizations including:

• autoconversion, accretion, self-collection, breakup, mean terminal velocity of raindrops, and rain evaporation rates from Seifert and Beheng 2006.
• number concentration adjustment from Horn 2012.
• additional double-moment bulk microphysics autoconversion and accretion rates from: Khairoutdinov and Kogan 2000, Beheng 1994, Tripoli and Cotton 1980, and Liu and Daum 2004.

pdf_cloud_parameters(pdf_c, q, ρₐ, N)

Return the parameters of the size distribution of cloud particles in terms of diameter.

The size distribution is given by:

n(D) = N₀c * D^νcD * exp(-λc * D^μcD)

where

• νcD = 3νc + 2
• μcD = 3μc

Arguments

• pdf_c: Size distribution parameters for cloud droplets, CMP.CloudParticlePDF_SB2006
• q: Liquid mass content [kg/kg]
• ρₐ: Air density [kg/m³]
• N: Number concentration of the particle [1/m³]

Returns

• (logN₀c, λc, νcD, μcD): Parameters of the generalized gamma distribution in terms of diameter

pdf_rain_parameters(pdf_r, qᵣ, ρₐ, Nᵣ)

Return the parameters of the rain drop diameter distribution

n_r(D) = N_0 * exp(- D / Dr_mean)

where

• D is the diameter of the raindrop,
• N_0 [1/m³] is the number concentration of raindrops,
• Dr_mean [m] is the mean diameter of the raindrops.

Note: in SB2006, Eq. (83) the distribution is given as:

f(D) = N_0 * exp(- λ_r D)

where λ_r ≡ 1 / Dr_mean [1/m] is the inverse of the mean diameter of the raindrops.

Arguments

• pdf_r: struct containing size distribution parameters for rain. Can either be CMP.RainParticlePDF_SB2006_notlimited or CMP.RainParticlePDF_SB2006_limited. For the latter, the values for N_0, Dr_mean, and xr_mean are limited to be within provided ranges.
• qᵣ: mass of rain water [kg]
• ρₐ: air density [kg/m³]
• Nᵣ: number of rain drops [1/m³]

Returns

• A NamedTuple with the fields (; N₀r, Dr_mean, xr_mean)

pdf_cloud_parameters_mass(pdf_c, q, ρₐ, N)

Return the parameters of the size distribution of cloud particles in terms of mass.

See log_pdf_cloud_parameters_mass for more details.

pdf_rain_parameters_mass(pdf_r::CMP.RainParticlePDF_SB2006, qᵣ, ρₐ, Nᵣ)

Return the parameters of the rain drop diameter distribution in terms of mass.

As a function of diameter, the size distribution is given by:

n(D) = N₀r * exp(-D / Dr_mean)

In terms of mass (x), the size distribution is given by:

f(x) = n(D(x)) * ∂D∂x(x)
     = N₀ * exp(-D(x) / Dr_mean) * 2 / (π * ρw) * x^(-2/3)
     = N₀ * 2 / (π * ρw) * x^(-2/3) * exp(- (6 / (π * ρw))^(1/3) / Dr_mean * x^(1/3))

where

• D(x) = (6x / (π * ρw))^(1/3) is the diameter of a raindrop of mass x.
• ∂D∂x(x) = (6 / (π * ρw))^(1/3) * x^(-2/3) is the derivative of the diameter with respect to the mass.

If we write the general form of the size distribution as:

f(x) = A * x^ν * exp(-B * x^μ)

then we have that:

• A = N₀ * 2 / (π * ρw)
• B = (6 / (π * ρw))^(1/3) / Dr_mean
• ν = -2/3
• μ = 1/3

log_pdf_cloud_parameters_mass(pdf_c, q, ρₐ, N)

Return the log of the parameters of the generalized gamma distribution of the form

f(x) = A * x^ν * exp(-B * x^μ),  [Eq. (79) in Seifert and Beheng 2006, but using the symbol `B` instead of `λ`]

where

B = [  x̄ Γ(z₁) / Γ(z₂) ]^(-μ)
A = μ N B^(z₁) / Γ(z₁)
z₁ = (ν + 1) / μ
z₂ = (ν + 2) / μ

That is,

log(B) = - μ [ log(x̄) + logΓ(z₁) - logΓ(z₂) ]
log(A) = log(μ) + log(N) + z₁ * log(B) - logΓ(z₁)

Arguments

• pdf_c: Size distribution parameters for cloud droplets, CMP.CloudParticlePDF_SB2006
• q: Liquid mass content [kg/kg]
• ρₐ: Air density [kg/m³]
• N: Number concentration of the particle [1/m³]

Returns

• (logA, logB): Log of the parameters of the generalized gamma distribution

size_distribution(pdf::CMP.RainParticlePDF_SB2006, q, ρₐ, N)

Return n(D), a function that computes the size distribution for rain particles at diameter D

Arguments

• pdf: Rain size distribution parameters, CMP.RainParticlePDF_SB2006
• q: Rain water specific content [kg/kg]
• ρₐ: Density of air [kg/m³]
• N: Rain water number concentration [1/m³]

size_distribution(pdf::CMP.CloudParticlePDF_SB2006, q, ρₐ, N)

Return n(D), a function that computes the size distribution for cloud particles at diameter D

The size distribution is given by:

n(D) = N₀c * D^(3νc + 2) * exp(-λc * D^(3μc))

Arguments

• pdf: Cloud size distribution parameters, CMP.CloudParticlePDF_SB2006
• q: Cloud water specific content [kg/kg]
• ρₐ: Density of air [kg/m³]
• N: Cloud water number concentration [1/m³]

size_distribution_value(pdf, q, ρₐ, N, D)

Return the size distribution value for a cloud or rain particle of diameter D.

See size_distribution for more details.

get_size_distribution_bounds(pdf, q, ρₐ, N, p)

Return the minimum and maximum diameters of a cloud or rain particle such that the size distribution is within (1 - p) to (p) probability of the true size distribution.

Arguments

• pdf: Size distribution parameters for cloud or rain, CMP.RainParticlePDF_SB2006 or CMP.CloudParticlePDF_SB2006
• q: mass mixing ratio of cloud or rain water
• ρₐ: density of air
• N: number mixing ratio of cloud or rain
• p: probability level (0 ≤ p ≤ 1)

Returns

• D_min, D_max: minimum and maximum diameters of a cloud or rain particle such that the size distribution is within (1 - p) to (p) probability of the true size distribution. All inputs and output diameters are in base SI units. The bounds are calculated through quantile functions of the size distribution.

A structure containing the rates of change of the specific contents and number densities of cloud liquid water and rain water.

autoconversion(acnv, pdf_c, q_lcl, q_rai, ρ, N_lcl)

Compute autoconversion rates

Arguments

• acnv: Autoconversion parameters, CMP.AcnvSB2006
• pdf_c: Cloud size distribution parameters, CMP.CloudParticlePDF_SB2006
• q_lcl: Cloud liquid water specific content [kg/kg]
• q_rai: Rain water specific content [kg/kg]
• ρ: Air density [kg/m³]
• N_lcl: Cloud droplet number density [1/m³]

Returns

• LclRaiRates with q_lcl, N_lcl, q_rai, N_rai tendencies due to collisions between cloud droplets (autoconversion)

accretion(accr, q_lcl, q_rai, ρ, N_lcl)

Compute accretion rate

Arguments

• accr: Accretion parameters, CMP.AccrSB2006
• q_lcl: Cloud liquid water specific content [kg/kg]
• q_rai: Rain water specific content [kg/kg]
• ρ: Air density [kg/m³]
• N_lcl: Cloud droplet number density [1/m³]

Returns

• LclRaiRates with q_lcl, N_lcl, q_rai, N_rai tendencies due to collisions between raindrops and cloud droplets (accretion)

accretion(accretion_scheme, q_lcl, q_rai, ρ)

• accretion_scheme - type for 2-moment rain accretion parameterization
• q_lcl - cloud liquid water specific content
• q_rai - rain water specific content
• ρ - air density (for KK2000Type and Beheng1994Type)

Returns the accretion rate of rain, parametrized following

• Khairoutdinov and Kogan (2000) for scheme == KK2000Type
• Beheng (1994) for scheme == B1994Type
• Tripoli and Cotton (1980) for scheme == TC1980Type

cloud_liquid_self_collection(acnv, pdf_c, q_lcl, ρ, dN_lcl_dt_au)

Compute cloud liquid self-collection rate

Arguments

• acnv: 2-moment autoconversion parameterization, CMP.AcnvSB2006
• pdf_c: Cloud size distribution parameters, CMP.CloudParticlePDF_SB2006
• q_lcl: Cloud liquid water specific content [kg/kg]
• ρ: Air density [kg/m³]
• dN_lcl_dt_au: Rate of change of cloud droplets number density due to autoconversion [1/m³/s]

Returns

• The cloud droplets number density tendency due to collisions of cloud droplets that produce larger cloud droplets (self-collection)

autoconversion_and_cloud_liquid_self_collection(scheme, q_lcl, q_rai, ρ, N_lcl)

Compute autoconversion and cloud liquid self-collection rates

Arguments

• scheme: 2-moment rain autoconversion parameterization, CMP.SB2006
• q_lcl: Cloud liquid water specific content [kg/kg]
• q_rai: Rain water specific content [kg/kg]
• ρ: Air density [kg/m³]
• N_lcl: Cloud droplet number density [1/m³]

Returns

• (au, sc): A NamedTuple containing the autoconversion rate and the cloud liquid self-collection rate.

rain_self_collection(pdf, self, q_rai, ρ, N_rai)

Compute the rain self-collection rate

Arguments

• pdf: Rain size distribution parameters, CMP.RainParticlePDF_SB2006
• self: Rain self-collection parameters, CMP.SelfColSB2006
• q_rai: Rain water specific content [kg/kg]
• ρ: Air density [kg/m³]
• N_rai: Raindrops number density [1/m³]

Returns

• The raindrops number density tendency due to collisions of raindrops that produce larger raindrops (self-collection).

rain_breakup(pdf, brek, q_rai, ρ, N_rai, dN_rai_dt_sc)

Compute the raindrops number density tendency due to breakup of raindrops

Arguments

• pdf: Rain size distribution parameters, CMP.RainParticlePDF_SB2006
• brek: Rain breakup parameters, CMP.BreakupSB2006
• q_rai: Rain water specific content
• ρ: Air density
• N_rai: Raindrops number density
• dN_rai_dt_sc: Rate of change of raindrops number density due to self-collection

Returns

• The raindrops number density tendency due to breakup of raindrops that produce smaller raindrops

rain_self_collection_and_breakup(params, q_rai, ρ, N_rai)

Compute the raindrops self-collection and breakup rates.

Arguments

• params: 2-moment rain size distribution parameters, CMP.SB2006 including raindrop size distribution, self collection, and breakup parameters
• q_rai: Rain water specific content
• ρ: Air density
• N_rai: Raindrops number density

Returns

• (sc, br): A NamedTuple containing the raindrops self-collection and breakup rates, respectively.

rain_terminal_velocity(SB2006, vel, q_rai, ρ, N_rai)

Compute the raindrops terminal velocity.

Arguments

• pdf_r: Rain size distribution parameters, CMP.RainParticlePDF_SB2006
• vel: Terminal velocity parameters, CMP.Chen2022VelTypeRain
• q_rai: Rain water specific content
• ρ: Air density
• N_rai: Raindrops number density

Returns

A tuple containing the number and mass weigthed mean fall velocities of raindrops in [m/s]. Assuming an exponential size distribution from Seifert and Beheng 2006 for scheme == SB2006Type Fall velocity of individual rain drops is parameterized:

• assuming an empirical relation similar to Rogers (1993) for velo_scheme == SB2006VelType
• following Chen et. al 2022, DOI: 10.1016/j.atmosres.2022.106171 for velo_scheme == Chen2022Type

rain_evaporation(evap, aps, tps, q_tot, q_lcl, q_icl, q_rai, q_sno, ρ, N_rai, T)

• evap - evaporation parameterization scheme
• aps - air properties
• tps - thermodynamics parameters
• q_tot, q_lcl, q_icl, q_rai, q_sno - total water, cloud liquid water, cloud ice, rain and snow specific contents,
• ρ - air density
• N_rai - raindrops number density
• T - air temperature

Returns a named tuple containing the tendency of raindrops number density and rain water specific content due to rain rain_evaporation, assuming a power law velocity relation for fall velocity of individual drops and an exponential size distribution, for scheme == SB2006Type

conv_q_lcl_to_q_rai(acnv, q_lcl, ρ, N_d; smooth_transition)

• acnv - 2-moment rain autoconversion parameterization
• q_lcl - cloud liquid water specific content
• ρ - air density
• N_d - prescribed cloud droplet number concentration

Returns the q_rai tendency due to collisions between cloud droplets (autoconversion), parametrized following:

• Khairoutdinov and Kogan (2000) for scheme == KK2000Type
• Beheng (1994) for scheme == B1994Type
• Tripoli and Cotton (1980) for scheme == TC1980Type
• Liu and Daum (2004) for scheme ==LD2004Type

The Beheng1994Type, TC1980Type and LD2004Type of schemes additionally accept smooth_transition flag that smoothes their thershold behaviour if set to true. The default value is false.

number_increase_for_mass_limit(numadj, x_max, q, ρ, N)

Compute the tendency (rate of change) of number concentration N required to ensure that the mean particle mass x = ρq / N does not exceed the upper limit x_max. Returns a positive tendency when the mean mass is too high (x > x_max), and zero otherwise. The method is based on Horn (2012, DOI: 10.5194/gmd-5-345-2012).

Arguments

• numadj: Number concentration adjustment parameters (CMP.NumberAdjustmentHorn2012)
• q: Mass mixing ratio [kg/kg]
• ρ: Air density [kg/m³]
• N: Number concentration [1/m³]
• x_max: Maximum allowed mean particle mass [kg]

Returns

• The rate of change of number concentration [1/(m³·s)] needed to bring the mean mass within the upper bound.

number_decrease_for_mass_limit(numadj, x_min, q, ρ, N)

Compute the tendency (rate of change) of number concentration N required to ensure that the mean particle mass x = ρq / N does not fall below the lower limit x_min. Returns a negative tendency when the mean mass is too low (x < x_min), and zero otherwise. The method is based on Horn (2012, DOI: 10.5194/gmd-5-345-2012).

Arguments

• numadj: Number concentration adjustment parameters (CMP.NumberAdjustmentHorn2012)
• q: Mass mixing ratio [kg/kg]
• ρ: Air density [kg/m³]
• N: Number concentration [1/m³]
• x_min: Minimum allowed mean particle mass [kg]

Returns

• The rate of change of number concentration [1/(m³·s)] needed to bring the mean mass within the lower bound.

DistributionTools

A module containing tools for working with size distributions.

Currently, it contains tools for working with the generalized gamma distribution and the exponential distribution.

Mostly used for the 2-moment microphysics.

generalized_gamma_quantile(ν, μ, B, Y)

Calculate the quantile (inverse cumulative distribution function) for a generalized gamma distribution parameterized in the form:

g(x) = A ⋅ x^ν ⋅ exp(-B ⋅ x^μ)

Arguments

• ν, μ, B: The PDF parameters
• Y: The probability level (0 ≤ Y ≤ 1) for which to compute the quantile

Returns

• x: The value x such that P(X ≤ x) = Y

generalized_gamma_cdf(ν, μ, B, x)

Calculate the cumulative distribution function (CDF) for a generalized gamma distribution parameterized in the form:

g(x) = A * x^ν * exp(-B * x^μ)

The CDF gives the probability P(X ≤ x).

Arguments

• ν, μ, B: The PDF parameters
• x: The value at which to evaluate the CDF

Returns

• p: The probability P(X ≤ x)

exponentialcdf(Dmean, D)

Calculate the cumulative distribution function (CDF) for an exponential distribution parameterized in the form:

n(D) = N₀ * exp(-D / D_mean)

where N₀ is a normalizing constant such that the total probability is 1.

Arguments

• D_mean: The mean value of the distribution
• D: The point at which to evaluate the CDF (must be ≥ 0)

Returns

• p: The probability P(X ≤ D)

exponential_quantile(D_mean, Y)

Calculate the quantile (inverse cumulative distribution function) for an exponential distribution parameterized in the form:

n(D) = N₀ * exp(-D / D_mean)

where N₀ is a normalizing constant such that the total probability is 1.

Arguments

• Y: The probability level (0 ≤ Y ≤ 1) for which to compute the quantile
• D_mean: The mean value of the distribution

Returns

• D: The value D such that P(X ≤ D) = Y

Predicted particle properties scheme P3 for ice [23].

See online docs for more information.

ParametersP3{FT}

Parameters for P3 bulk microphysics scheme.

From Morrison and Milbrandt (2015) [23]

Fields

• mass: Mass-size relation, e.g. MassPowerLaw

• area: Area-size relation, e.g. AreaPowerLaw

• slope: Slope relation, e.g. SlopePowerLaw or SlopeConstant

• vent: Ventilation relation, e.g. VentilationFactor

• ρ_rim_local: Local rime density, e.g. LocalRimeDensity

• τ_wet: Wet growth time scale [s]

• ρ_i: Cloud ice density [kg m⁻³]

• ρ_l: Cloud liquid water density [kg m⁻³]

• T_freeze: Water freeze temperature [K]

ParametersP3(FT)

Create a ParametersP3 object from a floating point type FT.

Examples

julia> import CloudMicrophysics.Parameters as CMP

julia> CMP.ParametersP3(Float64)
ParametersP3{Float64}
├── mass: MassPowerLaw
│   ├── α_va = 0.018537721864540644 [kg μm^(-β_va)]
│   └── β_va = 1.9 [-]
├── area: AreaPowerLaw
│   ├── γ = 0.2285 [μm^(2-σ)]
│   └── σ = 1.88 [-]
├── slope: SlopePowerLaw
│   ├── a = 0.00191 [m^b]
│   ├── b = 0.8 [-]
│   ├── c = 2.0 [-]
│   └── μ_max = 6.0 [-]
├── vent: VentilationFactor
│   ├── aᵥ = 0.78 [-]
│   └── bᵥ = 0.308 [-]
├── ρ_rim_local: LocalRimeDensity
│   ├── a = 51.0 [m^b]
│   ├── b = 114.0 [-]
│   ├── c = -5.5 [-]
│   └── ρ_ice = 916.7 [-]
├── τ_wet = 100.0 [s]
├── ρ_i = 916.7 [kg m⁻³]
├── ρ_l = 1000.0 [kg m⁻³]
└── T_freeze = 273.15 [K]

ParametersP3(toml_dict::CP.ParamDict; [slope_law = :powerlaw])

Create a ParametersP3 object from a ClimaParams TOML dictionary.

Arguments

• toml_dict::CP.ParamDict: A ClimaParams TOML dictionary
• slope_law: Slope law to use (:constant or, by default, :powerlaw)

MassPowerLaw{FT}

Parameters for mass(size) relation.

From measurements of mass grown by vapor diffusion and aggregation in midlatitude cirrus by Brown and Francis (1995) [24]

A part of the ParametersP3 parameter set.

Fields

• α_va: Coefficient in mass(size) relation [kg m^(-β_va)]

• β_va: Coefficient in mass(size) relation [-]

AreaPowerLaw{FT}

Parameters for area(size) relation.

\[A(D) = γ D^σ\]

where γ and σ are coefficients in area(size) for ice side plane, column, bullet, and planar polycrystal aggregates. Values are from Mitchell (1996) [25]

A part of the ParametersP3 parameter set.

Fields

• γ: Scale [μm^(2-σ)]

• σ: Power [-]

SlopeLaw{FT}

The top-level super-type for slope parameterizations.

See SlopePowerLaw and SlopeConstant for concrete implementations.

SlopePowerLaw{FT}

Slope parameter μ as a power law in shape parameter λ:

\[μ(λ) = a λ^b - c\]

and is limited to:

\[0 ≤ μ ≤ μ_{max}\]

See also Eq. 3 in Morrison and Milbrandt (2015) [23]

A part of the ParametersP3 parameter set.

Fields

• a: Scale [m^b]

• b: Power [-]

• c: Offset [-]

• μ_max: Upper limiter [-]

SlopeConstant{FT}

Slope parameter μ as a constant:

\[μ(λ) = μ_{const}\]

A part of the ParametersP3 parameter set.

Fields

• μ: Slope parameter μ [-]

VentilationFactor{FT}

Parameters for ventilation factor:

\[F(D) = a_{v} + b_{v} N_{Sc}^{1/3} N_{Re}(D)^{1/2}\]

where N_{Sc} is the Schmidt number and N_{Re}(D) is the Reynolds number for a particle with diameter D.

From Seifert and Beheng (2006) [10], see also Eq. (13-61) in Pruppacher and Klett (2010) [65]

A part of the ParametersP3 parameter set.

Fields

• aᵥ: Constant coefficient in ventilation factor [-]

• bᵥ: Linear coefficient in ventilation factor [-]

LocalRimeDensity{FT}
(ρ′_rim::LocalRimeDensity)(Rᵢ)

Local rime density parameterization based on Cober and List (1993) [32], Eq. 16 and 17.

Given an instance ρ′_rim::LocalRimeDensity, obtain the local rime density for a given Rᵢ [m² s⁻¹ °C⁻¹] by calling ρ′_rim(Rᵢ).

The parameterization is given by:

\[ρ'_{rim} = a + b R_i + c R_i^2, \quad 1 ≤ R_i ≤ 8,\]

The range is extended to R_i ≤ 12, by linearly interpolating between ρ′_rim(8) and ρ_ice = 900 kg/m³. The latter is the solid bulk ice density.

For calculating Rᵢ, see compute_local_rime_density.

P3State{FT}

State of the P3 scheme.

This struct bundles the P3 parameterizations params, the provided rime state (F_rim, ρ_rim), and the derived threshold variables (D_th, D_gr, D_cr, ρ_g).

To obtain a P3State object, use the get_state function.

Fields

• params: CMP.ParametersP3 object

• L_ice: The ice mass concentration [kg/m³]

• N_ice: The ice number concentration [1/m³]

• F_rim: Rime mass fraction

• ρ_rim: Rime density

get_state(params; L_ice, N_ice, F_rim, ρ_rim)

Create a P3State from CMP.ParametersP3 and rime state parameters.

Arguments

• params: CMP.ParametersP3 object

Keyword arguments

• L_ice: ice mass concentration [kg/m³]
• N_ice: ice number concentration [1/m³]
• F_rim: rime mass fraction [-], F_rim = L_rim / L_ice
• ρ_rim: rime density [kg/m³], ρ_rim = L_rim / B_rim

Examples

```jldoctest julia> import CloudMicrophysics.Parameters as CMP, CloudMicrophysics.P3Scheme as P3

julia> FT = Float32;

julia> params = CMP.ParametersP3(FT);

julia> state = P3.getstate(params; Frim = FT(0.5), ρrim = FT(916.7)) P3State{Float32} ├── params = {MassPowerLaw, AreaPowerLaw, SlopePowerLaw, VentilationFactor} ├── Frim = 0.5 [-] └── ρ_rim = 916.7 [kg/m^3] ```

get_thresholds_ρ_g(state::P3State)
get_thresholds_ρ_g(params::CMP.ParametersP3, F_rim, ρ_rim)

Returns

• (; D_th, D_gr, D_cr, ρ_g): The thresholds for the size distribution, and the density of total (deposition + rime) ice mass for graupel [kg/m³]

See get_D_th, get_D_gr, get_D_cr, and get_ρ_g for more details.

get_ρ_d(mass::MassPowerLaw, F_rim, ρ_rim)
get_ρ_d(state::P3State)

Exact solution for the density of the unrimed portion of the particle as function of the rime mass fraction F_rim, mass power law parameters mass, and rime density ρ_rim.

Arguments

• mass: CMP.MassPowerLaw parameters
• F_rim: rime mass fraction
• ρ_rim: rime density

Returns

• ρ_d: density of the unrimed portion of the particle [kg/m³]

Examples

julia> import CloudMicrophysics.Parameters as CMP,
              ClimaParams as CP,
              CloudMicrophysics.P3Scheme as P3

julia> FT = Float64;

julia> mass = CMP.MassPowerLaw(CP.create_toml_dict(FT));

julia> F_rim, ρ_rim = FT(0.5), FT(916.7);

julia> ρ_d = P3.get_ρ_d(mass, F_rim, ρ_rim)
488.9120789986412

get_ρ_g(F_rim, ρ_rim, ρ_d)
get_ρ_g(mass::MassPowerLaw, F_rim, ρ_rim)

Return the density of total (deposition + rime) ice mass for graupel [kg/m³]

Arguments

• F_rim: rime mass fraction (L_rim / L_ice) [-]
• ρ_rim: rime density (L_rim / B_rim) [kg/m³]
• ρ_d: density of the unrimed portion of the particle [kg/m³], see get_ρ_d

Returns

• ρ_g: density of total (deposition + rime) ice mass for graupel [kg/m³]

Notes:

See Eq. 16 in [23].

get_D_th(mass::MassPowerLaw, ρ_i)

Return the critical size separating spherical and nonspherical ice [meters]

See Eq. 8 in [23].

get_D_gr(mass::MassPowerLaw, ρ_g)

Return the size of equal mass for graupel and unrimed ice [meters]

See Eq. 15 in [23].

get_D_cr(mass::MassPowerLaw, F_rim, ρ_g)

Return the size of equal mass for graupel and partially rimed ice [meters]

See Eq. 14 in [23].

ice_mass(state, D)
ice_mass(params, F_rim, ρ_rim, D)

Return the mass of a particle based on where it falls in the particle-size-based properties regime.

Arguments

• state: The P3State, or
• params, F_rim, ρ_rim: The CMP.ParametersP3, rime mass fraction, and rime density,
• D: maximum particle dimension [m]

ice_density(state::P3State, D)
ice_density(params::CMP.ParametersP3, F_rim, ρ_rim, D)

Return the density of a particle at diameter D

Arguments

• state: The P3State, or
• params, F_rim, ρ_rim: The CMP.ParametersP3, rime mass fraction, and rime density,
• D: maximum particle dimension [m]

Notes:

The density of nonspherical particles is assumed to be the particle mass divided by the volume of a sphere with the same D [23]. Needed for aspect ratio calculation, so we assume zero liquid fraction.

∂ice_mass_∂D(state::P3State, D)
∂ice_mass_∂D(params::CMP.ParametersP3, F_rim, ρ_rim, D)

Return the derivative of the ice mass with respect to the particle diameter.

ice_area(state::P3State, D)
ice_area(params::CMP.ParametersP3, F_rim, ρ_rim, D)

Return the cross-sectional area of a particle based on where it falls in the particle-size-based properties regime.

Arguments

• state: P3State object, or
• params, F_rim, ρ_rim: The CMP.ParametersP3, rime mass fraction, and rime density,
• D: maximum particle dimension [m]

ϕᵢ(state::P3State, D)
ϕᵢ(params::CMP.ParametersP3, F_rim, ρ_rim, D)

Returns the aspect ratio (ϕ) for an ice particle

Arguments

• state: The P3State, or
• params, F_rim, ρ_rim: The CMP.ParametersP3, rime mass fraction, and rime density,
• D: maximum dimension of ice particle [m]

Notes

The density of nonspherical particles is assumed to be equal to the particle mass divided by the volume of a spherical particle with the same D_max [23]. Assuming zero liquid fraction and oblate shape.

get_μ(slope::CMP.SlopeLaw, logλ)
get_μ(state::P3State, logλ)

Compute the slope parameter μ

Arguments

• slope: CMP.SlopeLaw object, or
• state: P3State object, or
• params: CMP.ParametersP3 object
• logλ: The log of the slope parameter [log(1/m)]

get_logN₀(N_ice, μ, logλ)

Compute log(N₀) given the state, N, and logλ,

    N  = N₀ ∫ G(D) dD
log N₀ = log N - log(∫G(D) dD) 
       = log(N) - log( ∫D^μ e^{-λD} dD )
       = log(N) - M⁰

Arguments

• N_ice: The number concentration [1/m³]
• μ: The shape parameter [-]
• logλ: The log of the slope parameter [log(1/m)]

get_distribution_logλ(params, L_ice, N_ice, F_rim, ρ_rim; [logλ_min, logλ_max])
get_distribution_logλ(state; [logλ_min, logλ_max])

Solve for the distribution parameters given the state, and the mass (L) and number (N) concentrations.

The assumed distribution is of the form

\[N′(D) = N₀ D^μ e^{-λD}\]

where N′(D) is the number concentration at diameter D and μ is the slope parameter. The slope parameter is parameterized, e.g. CMP.SlopePowerLaw or CMP.SlopeConstant.

This algorithm solves for logλ = log(λ) and log_N₀ = log(N₀) given L_ice and N_ice by solving the equations:

\[\begin{align*} \log(L) &= \log ∫_0^∞ m(D) N′(D)\ \mathrm{d}D, \\ \log(N) &= \log ∫_0^∞ N′(D)\ \mathrm{d}D, \\ \end{align*}\]

where m(D) is the mass of a particle at diameter D (see ice_mass). The procedure is decribed in detail in the P3 docs.

Arguments

• state: P3State object
• params: CMP.ParametersP3 object
• L_ice: The mass concentration [kg/m³]
• N_ice: The number concentration [1/m³]
• F_rim: The rime mass fraction
• ρ_rim: The rime density

Keyword arguments

• logλ_min: The minimum value of the search bounds [log(1/m)], default is log(1e1)
• logλ_max: The maximum value of the search bounds [log(1/m)], default is log(1e7)

logN′ice(state, logλ)

Compute the log of the ice particle number concentration at diameter D given the distribution dist

size_distribution(state::P3State, logλ)

Return n(D), a function that computes the size distribution for ice particles at diameter D

Arguments

• state: The P3State
• logλ: The log of the slope parameter [log(1/m)]

loggamma_inc_moment(D₁, D₂, μ, logλ, [k = 0], [scale = 1])

Compute log(Iᵏ) where Iᵏ is the following integral:

``I^k = ∫_{D₁}^{D₂} G(D) D^k dD``

$G(D) ≡ D^μ e^{-λD}$ is the (unnormalized) gamma kernel, and k is an arbitrary exponent.

If scale is provided, log(scale ⋅ Iᵏ) is returned.

With appropriate scaling, we can compute useful quantities like:

• the k-th moment of the ice PSD, $M^k = N₀ I^k$
• combined power law and moment weighted integrals, $∫_{D₁}^{D₂} (aD^b) D^n K(D) dD ≡ a I^(b + n)$

Arguments

• D₁: The minimum diameter [m]
• D₂: The maximum diameter [m]
• μ: The PSD shape parameter [-]
• logλ: The log of the PSD slope parameter [log(1/m)]
• k: An arbitrary exponent [-], default is 0
• scale: The scale factor [-], default is 1

Extended help

Implementation details

We can write ∫_D₁^D₂ G(D) D^k dD, where G(D) = D^μ e^{-λD} as: ∫_D₁^∞ G(D) D^k dD - ∫_D₂^∞ G(D) D^k dD with the transformation x = λD, and z = μ+k+1, each term can be written as: ∫_{Dᵢ}^∞ G(D) D^k dD = ∫_{λDᵢ}^∞ x^z e^{-x} dx / λ^z = Γ(z, λDᵢ) / λ^z where Γ(z, λDᵢ) = q ⋅ Γ(z) and q is the incomplete gamma function ratio given by (_, q) = SF.gamma_inc(z, x). This means that the integral ∫_{Dᵢ}^∞ G(D) D^k dD is computed as: Γ(z) ⋅ q / λ^z The full integral from D₁ to D₂ is then: Γ(z) ⋅ (q_D₁ - q_D₂) / λ^z In log-space, this is: - z log(λ) + logΓ(z) + log(q_D₁ - q_D₂)

loggamma_moment(μ, logλ; [k = 0], [scale = 1])

Compute log(scale ⋅ ∫_0^∞ G(D) D^k dD), where G(D) ≡ D^μ e^{-λD} is the (unnormalized) gamma kernel, k is an arbitrary exponent, and scale is a scale factor.

Arguments

• μ: The PSD shape parameter [-]
• logλ: The log of the PSD slope parameter [log(1/m)]

Keyword arguments

• k: An arbitrary exponent [-], default is 0
• scale: The scale factor [-], default is 1.

The implementation follows the same logic as loggamma_inc_moment, but with D₁ = 0 and D₂ = ∞, which implies q_D₁ = 1 and q_D₂ = 0.

logmass_gamma_moment(state, logλ; [n=0])

Compute log(∫_0^∞ Dⁿ m(D) N′(D) dD) given the state and logλ. This is the log of the n-th moment of the mass-weighted PSD.

Arguments

• state: P3State object
• μ: The shape parameter [-]
• logλ: The log of the slope parameter [log(1/m)]

Keyword arguments

• n: The order of the moment, default is 0

Note:

• For n = 0, this evaluates to log(L/N₀)
• For n = 1, this evaluates to the (unnormalized) mass-weighted mean particle size, see D_m

logLdivN(state, logλ)

Compute log(L/N) given the state and logλ

Arguments

• state: P3State object
• logλ: The log of the slope parameter [log(1/m)]

D_m(state::P3State, logλ)

Compute the mass weighted mean particle size [m]

Parameters

• state: P3State object
• logλ: The log of the slope parameter [log(1/m)]

ice_particle_terminal_velocity(velocity_params, ρₐ, state::P3State; [use_aspect_ratio])
ice_particle_terminal_velocity(velocity_params, ρₐ, params::CMP.ParametersP3, F_rim, ρ_rim; [use_aspect_ratio])

Returns the terminal velocity of a single ice particle as a function of its size (maximum dimension, D) using the Chen 2022 parametrization.

Arguments

• state: A P3State
• velocity_params: A CMP.Chen2022VelType with terminal velocity parameters
• ρₐ: Air density [kg/m³]

Keyword arguments

• use_aspect_ratio: Bool flag set to true if we want to consider the effects of particle aspect ratio on its terminal velocity (default: true)

ice_terminal_velocity_number_weighted(
    velocity_params::CMP.Chen2022VelType, ρₐ, state::P3State, logλ;
    [use_aspect_ratio], [∫kwargs...],
)

Return the terminal velocity of the number-weighted mean ice particle size.

Arguments

• velocity_params: A CMP.Chen2022VelType with terminal velocity parameters
• ρₐ: Air density [kg/m³]
• state: A P3State
• logλ: The log of the slope parameter [log(1/m)]

Keyword arguments

• use_aspect_ratio: Bool flag set to true if we want to consider the effects of particle aspect ratio on its terminal velocity (default: true)
• ∫kwargs...: Additional optional keyword arguments to pass to the quadrature rule

See also ice_terminal_velocity_mass_weighted

ice_terminal_velocity_mass_weighted(velocity_params::CMP.Chen2022VelType, ρₐ, state::P3State, logλ; [use_aspect_ratio], [∫kwargs...])

Return the terminal velocity of the mass-weighted mean ice particle size.

Arguments

• velocity_params: A CMP.Chen2022VelType with terminal velocity parameters
• ρₐ: Air density [kg/m³]
• state: A P3State
• logλ: The log of the slope parameter [log(1/m)]

Keyword arguments

• use_aspect_ratio: Bool flag set to true if we want to consider the effects of particle aspect ratio on its terminal velocity (default: true)
• ∫kwargs...: Keyword arguments passed to the quadrature rule

See also ice_terminal_velocity_number_weighted

het_ice_nucleation(pdf_c, p3, tps, q_lcl, N, T, ρₐ, p, aerosol)

• aerosol - aerosol parameters (supported types: desert dust, illite, kaolinite)
• tps - thermodynamics parameters
• q_lcl - cloud liquid water specific content
• N_lcl - cloud droplet number concentration
• RH - relative humidity
• T - temperature
• ρₐ - air density
• dt - model time step

Returns a named tuple with ice number concentration and ice content hetergoeneous freezing rates from cloud droplets.

ice_melt(velocity_params::CMP.Chen2022VelType, aps, tps, Tₐ, ρₐ, dt, state, logλ; ∫kwargs...)

Arguments

• velocity_params: CMP.Chen2022VelType
• aps: CMP.AirProperties
• tps: thermodynamics parameters
• Tₐ: temperature (K)
• ρₐ: air density
• dt: model time step (for limiting the tendnecy)
• state: a P3State object
• logλ: the log of the slope parameter [log(1/m)]

Keyword arguments

• ∫kwargs...: Additional keyword arguments passed to the quadrature rule

Returns the melting rate of ice (QIMLT in Morrison and Mildbrandt (2015)).

bulk_liquid_ice_collision_sources(
    params, logλ, L_ice, F_rim, ρ_rim,
    psd_c, psd_r, L_c, N_c, L_r, N_r,
    aps, tps, vel, ρₐ, T,
)

Computes the bulk rates for ice and liquid particle collisions.

Arguments

• params: the CMP.ParametersP3
• logλ: the log of the slope parameter [log(1/m)]
• L_ice: ice water content [kg/m³]
• F_rim: riming fraction
• ρ_rim: rime density [kg/m³]
• psd_c: a CMP.CloudParticlePDF_SB2006
• psd_r: a CMP.RainParticlePDF_SB2006
• L_c: cloud liquid water content [kg/m³]
• N_c: cloud liquid water number concentration [1/m³]
• L_r: rain water content [kg/m³]
• N_r: rain number concentration [1/m³]
• aps: CMP.AirProperties
• tps: thermodynamics parameters
• vel: the velocity parameterization, e.g. CMP.Chen2022VelType
• ρₐ: air density [kg/m³]
• T: temperature [K]

Returns

A NamedTuple of (; ∂ₜq_c, ∂ₜq_r, ∂ₜN_c, ∂ₜN_r, ∂ₜL_rim, ∂ₜL_ice, ∂ₜB_rim), where:

1. ∂ₜq_c: cloud liquid water content tendency [kg/kg/s]
2. ∂ₜq_r: rain water content tendency [kg/kg/s]
3. ∂ₜN_c: cloud number concentration tendency [1/m³/s]
4. ∂ₜN_r: rain number concentration tendency [1/m³/s]
5. ∂ₜL_rim: riming mass tendency [kg/m³/s]
6. ∂ₜL_ice: ice water content tendency [kg/m³/s]
7. ∂ₜB_rim: rime volume tendency [m³/m³/s]

volumetric_collision_rate_integrand(state, velocity_params, ρₐ)

Returns a function that computes the volumetric collision rate integrand for ice-liquid collisions [m³/s]. The returned function takes ice and liquid particle diameters as arguments.

Arguments

• state: P3State
• velocity_params: velocity parameterization, e.g. CMP.Chen2022VelType
• ρₐ: air density

Returns

A function (D_ice, D_liq) -> E * K * |vᵢ - vₗ| where:

• D_ice and D_liq are the (maximum) diameters of the ice and liquid particles
• E is the collision efficiency
• K is the collision cross section
• vᵢ and vₗ are the terminal velocities of ice and liquid particles

Note that E, K, vᵢ and vₗ are all, in general, functions of D_ice and D_liq.

This function is a component of integrals like

\[∫ ∫ E * K * |vᵢ - vₗ| * N'_i * N'_l dD_i dD_l\]

compute_max_freeze_rate(aps, tps, velocity_params, ρₐ, Tₐ, state)

Returns a function max_freeze_rate(Dᵢ) that returns the maximum possible freezing rate [kg/s] for an ice particle of diameter Dᵢ [m]. Evaluates to 0 if T ≥ T_freeze.

Arguments

• aps: CMP.AirProperties
• tps: TDP.ThermodynamicsParameters
• velocity_params: velocity parameterization, e.g. CMP.Chen2022VelType
• ρₐ: air density [kg/m³]
• Tₐ: air temperature [K]
• state: P3State

This rate represents the thermodynamic upper limit to collisional freezing, which occurs when the heat transfer from the ice particle to the environment is balanced by the latent heat of fusion.

From Eq (A7) in Musil (1970), [31].

compute_local_rime_density(velocity_params, ρₐ, T, state)

Provides a function ρ′_rim(Dᵢ, Dₗ) that computes the local rime density [kg/m³] for a given ice particle diameter Dᵢ [m] and liquid particle diameter Dₗ [m].

Arguments

• velocity_params: velocity parameterization, e.g. CMP.Chen2022VelType
• ρₐ: air density [kg/m³]
• T: temperature [K]
• state: P3State

Returns

A function that computes the local rime density [kg/m³] using the equation:

\[ρ'_{rim} = a + b R_i + c R_i^2\]

where

\[R_i = \frac{ 10^6 ⋅ D_{liq} ⋅ |v_{liq} - v_{ice}| }{ 2 T_{sfc} }\]

and $T_{sfc}$ is the surface temperature [°C], $D_{liq}$ is the liquid particle diameter [m], $v_{liq/ice}$ is the particle terminal velocity [m/s]. So the units of $R_i$ are [m² s⁻¹ °C⁻¹]. The units of $ρ'_{rim}$ are [kg/m³].

We assume for simplicity that $T_{sfc}$ equals $T$, the ambient air temperature. For real graupel, $T_{sfc}$ is slightly higher than $T$ due to latent heat release of freezing liquid particles onto the ice particle. Morrison & Milbrandt (2013) found little sensitivity to "realistic" increases in $T_{sfc}$.

See also LocalRimeDensity.

Extended help

Implementation follows Cober and List (1993), Eq. 16 and 17. See also the P3 fortran code, microphy_p3.f90, Line 3315-3323, which extends the range of the calculation to $R_i ≤ 12$, the upper limit of which then equals the solid bulk ice density, $ρ_ice = 916.7 kg/m^3$.

Note that Morrison & Milbrandt (2015) [23] only uses this parameterization for collisions with cloud droplets. For rain drops, they use a value near the solid bulk ice density, $ρ^* = 900 kg/m^3$. We do not consider this distinction, and use this parameterization for all liquid particles.

get_liquid_integrals(n, ∂ₜV, m_liq, ρ′_rim, liq_bounds; ∫kwargs...)

Returns a function liquid_integrals(Dᵢ) that computes the liquid particle integrals for a given ice particle diameter Dᵢ.

Arguments

• n: liquid particle size distribution function n(D)
• ∂ₜV: volumetric collision rate integrand function ∂ₜV(Dᵢ, D)
• m_liq: liquid particle mass function m_liq(D)
• ρ′_rim: local rime density function ρ′_rim(Dᵢ, D)
• liq_bounds: integration bounds for liquid particles

Keyword arguments

• ∫kwargs...: Additional keyword arguments passed to the quadrature rule

Notes

The function liquid_integrals(Dᵢ) returns a tuple (∂ₜN_col, ∂ₜM_col, ∂ₜB_col) of collision rates at Dᵢ, where:

• ∂ₜN_col: number collision rate [1/s]
• ∂ₜM_col: mass collision rate [kg/s]
• ∂ₜB_col: rime volume collision rate [m³/s]

∫liquid_ice_collisions(
    n_i, ∂ₜM_max, cloud_integrals, rain_integrals, ice_bounds; ∫kwargs...
)

Computes the bulk collision rate integrands between ice and liquid particles.

Arguments

• n_i: ice particle size distribution function n_i(D)
• ∂ₜM_max: maximum freezing rate function ∂ₜM_max(Dᵢ)
• cloud_integrals: an instance of get_liquid_integrals for cloud particles
• rain_integrals: an instance of get_liquid_integrals for rain particles
• ice_bounds: integration bounds for ice particles, from integral_bounds

Keyword arguments

• ∫kwargs...: Additional keyword arguments passed to the quadrature rule

Returns

A tuple of 8 integrands, see ∫liquid_ice_collisions for details.

∫liquid_ice_collisions(
    state, logλ, psd_c, psd_r, L_c, N_c, L_r, N_r, 
    aps, tps, vel, ρₐ, T, m_liq
)

Compute key liquid-ice collision rates and quantities. Used by bulk_liquid_ice_collision_sources.

Arguments

• state: P3State
• logλ: the log of the slope parameter [log(1/m)]
• psd_c: CMP.CloudParticlePDF_SB2006
• psd_r: CMP.RainParticlePDF_SB2006
• L_c: cloud liquid water content [kg/m³]
• N_c: cloud liquid water number concentration [1/m³]
• L_r: rain water content [kg/m³]
• N_r: rain number concentration [1/m³]
• aps: CMP.AirProperties
• tps: TDP.ThermodynamicsParameters
• vel: velocity parameterization, e.g. CMP.Chen2022VelType
• ρₐ: air density [kg/m³]
• T: temperature [K]
• m_liq: liquid particle mass function m_liq(D)

Returns

A tuple (QCFRZ, QCSHD, NCCOL, QRFRZ, QRSHD, NRCOL, ∫M_col, BCCOL, BRCOL, ∫𝟙_wet_M_col), where:

1. QCFRZ - Cloud mass collision rate due to freezing [kg/s]
2. QCSHD - Cloud mass collision rate due to shedding [kg/s]
3. NCCOL - Cloud number collision rate [1/s]
4. QRFRZ - Rain mass collision rate due to freezing [kg/s]
5. QRSHD - Rain mass collision rate due to shedding [kg/s]
6. NRCOL - Rain number collision rate [1/s]
7. ∫M_col - Total collision rate [kg/s]
8. BCCOL - Cloud rime volume source [m³/m³/s]
9. BRCOL - Rain rime volume source [m³/m³/s]
10. ∫𝟙_wet_M_col - Wet growth indicator [kg/s]

integrate(f, a, b, quad = ChebyshevGauss(100))

Approximate the definite integral ∫ₐᵇ f(x) dx using Chebyshev-Gauss quadrature of the first kind.

Mathematical Background

This method transforms the integral to the standard interval [-1, 1] and applies Chebyshev-Gauss quadrature. The transformation is:

 y = (2x - (a+b)) / (b-a)    →    x = (b-a)y/2 + (a+b)/2

with Jacobian dx/dy = (b-a)/2.

The integral becomes:

 ∫ₐᵇ f(x) dx = (b-a)/2 ∫₋₁¹ f(x(y)) dy

Using the Chebyshev-Gauss quadrature identity:

 ∫₋₁¹ g(y) dy = ∫₋₁¹ [g(y)√(1-y²)] / √(1-y²) dy ≈ (π/n) ∑ᵢ₌₁ⁿ g(yᵢ)√(1-yᵢ²)

where yᵢ = cos((2i-1)π/(2n)) are the Chebyshev nodes of the first kind.

Final Formula

 ∫ₐᵇ f(x) dx ≈ (b-a)π/(2n) ∑ᵢ₌₁ⁿ f(xᵢ)√(1-yᵢ²)

where:

• yᵢ = cos((2i-1)π/(2n)) for i = 1, ..., n
• xᵢ = (b-a)yᵢ/2 + (a+b)/2
• All quadrature weights are equal: π/n

Arguments

• f: Function to integrate
• a, b: Integration bounds. Note: if a ≥ b, or a or b is NaN, zero(f(a)) is returned.

Keyword arguments

• quad: Quadrature scheme, default: ChebyshevGauss(100)

Returns

Approximation to the definite integral ∫ₐᵇ f(x) dx

Notes

This method achieves spectral convergence for smooth functions and is particularly effective for analytic functions. The √(1-y²) weighting factor helps handle functions with mild singularities at the interval endpoints. Ref: https://en.wikipedia.org/wiki/Chebyshev–Gauss_quadrature

integrate(f, bnds...; quad = ChebyshevGauss(100))

Integrate the function f over each subinterval of the integration bounds, bnds.

Arguments

• f: Function to integrate
• bnds: A tuple of bounds, (a, b, c, d, ...)
• quad: Quadrature scheme, default: ChebyshevGauss(100)

The integral is computed as the sum of the integrals over each subinterval, (a, b), (b, c), (c, d), ....

ChebyshevGauss(n)

Quadrature scheme for Chebyshev-Gauss quadrature of the first kind.

Arguments

• n: Number of quadrature points

Available methods

node(::ChebyshevGauss, i::FT, n) where {FT}
weight(::ChebyshevGauss, i::FT, n) where {FT}
inv_weight_fun(::ChebyshevGauss, y)

• node(quad, i, n): Return the i-th node of the n-point Chebyshev-Gauss quadrature scheme.
• weight(quad, i, n): Return the i-th weight of the n-point Chebyshev-Gauss quadrature scheme.
• inv_weight_fun(quad, y): Return the inverse of the weight function w(x).

Mathematical Background

A method to approximate the value of integrals of the kind

∫_{-1}^{1} f(x) w(x) dx ≈ ∑_1^n f(x_i) wᵢ(x_i)

where w(x) = 1 / √(1 - x^2) is the weight function, x_i are the nodes, and wᵢ are the weights.

If we are interested in only the integral of f(x), we can instead integrate g(x) = f(x) / w(x),

∫_{-1}^{1} g(x) w(x) dx ≈ ∑_1^n g(x_i) wᵢ(x_i) = ∑_1^n f(x_i) wᵢ(x_i) / w(x_i)

References

• https://en.wikipedia.org/wiki/Chebyshev–Gauss_quadrature
• https://en.wikipedia.org/wiki/Gaussian_quadrature

integral_bounds(state::P3State, logλ; p, moment_order = 0)

Compute the integration bounds for the P3 size distribution,

Mⁿ = ∫_a^b Dⁿ * N′(D) dD = N₀ * ∫_a^b Dⁿ D^μ * exp(-λ * D) dD

where Mⁿ is the n-th moment of the size distribution. Here n ≡ moment_order and a and b are the integration bounds. For a proper moment, a=0 and b=∞. For the numerical integration, a and b are determined by this function.

Arguments

• state: P3State object
• logλ: The log of the slope parameter [log(1/m)]
• p: The integration bounds are set to the p-th and 1-p-th quantiles of the size distribution.
• moment_order: For integrands proportional to moments of the size distribution, moment_order can be used to indicate the order of the moment. May provide more accurate bounds; thus more accurate integration. Default: moment_order = 0.

Returns

• bnds: The integration bounds (a Tuple), for use in numerical integration (c.f. integrate).

A container for information on aerosol size distribution
and chemical properties.

The size distribution is a sum of lognormal internally mixed modes.
The chemical composition can be expressed using kappa parameter
or hygroscopicity parameter B.

Mode_B

Represents the sizes and chemical composition of aerosol particles in one size distribution mode. The mode is assumed to be made up of internally mixed components and follow a lognormal size distribution. The chemical composition of aerosol particles in this mode is described using the parameters from Abdul-Razzak and Ghan 2000.

Mode_κ

Represents the sizes and chemical composition of aerosol particles in one size distribution mode. The mode is assumed to be made up of internally mixed components and follow a lognormal size distribution. The chemical composition of aerosol particles in this mode is described using the parameters from Petters and Kreidenweis 2007.

AerosolDistribution

Represents the aerosol size distribution as a tuple with different modes. All modes have to either be of type ModeB (Abdul-Razzak and Ghan 2000) or of type Modeκ (Petters and Kreidenweis 2007).

Constructors

AerosolDistribution(modes::T)

Aerosol activation scheme, which includes:

• mean hygroscopicity for each mode of the aerosol size distribution
• critical supersaturation for each mode of the aerosol size distribution
• maximum supersaturation
• total number of particles actived
• total mass of particles actived

mean_hygroscopicity_parameter(ap, ad)

• ap - a struct with aerosol activation parameters
• ad - a struct with aerosol distribution (B or κ based)

Returns a tuple of hygroscopicity parameters (one tuple element for each aerosol size distribution mode). The tuple is computed either as mass-weighted B parameters (Abdul-Razzak and Ghan 2000) or volume weighted kappa parameters (Petters and Kreidenweis 2007). Implemented via a dispatch based on aerosol distribution mode type.

max_supersaturation(ap, ad, aip, tps, T, p, w, q_tot, q_liq, q_ice, N_liq, N_ice)

• ap - a struct with aerosol activation parameters
• ad - a struct with aerosol distribution
• aip - a struct with air parameters
• tps - a struct with thermodynamics parameters
• T - air temperature
• p - air pressure
• w - vertical velocity
• q_tot - total water specific content
• q_liq - liquid water specific content
• q_ice - ice water specific content
• N_liq - liquid water number concentration
• N_ice - ice water number concentration

Returns the maximum supersaturation.

N_activated_per_mode(ap, ad, aip, tps, T, p, w, q_tot, q_liq, q_ice, N_liq, N_ice)

• ap - a struct with aerosol activation parameters
• ad - aerosol distribution struct
• aip - a struct with air parameters
• tps - a struct with thermodynamics parameters
• T - air temperature
• p - air pressure
• w - vertical velocity
• q_tot - total water specific content
• q_liq - liquid water specific content
• q_ice - ice water specific content
• N_liq - liquid water number concentration
• N_ice - ice water number concentration

Returns the number of activated aerosol particles in each aerosol size distribution mode.

M_activated_per_mode(ap, ad, aip, tps, T, p, w, q_tot, q_liq, q_ice, N_liq, N_ice)

• ap - a struct with aerosol activation parameters
• ad - a struct with aerosol distribution parameters
• aip - a struct with air parameters
• tps - a struct with thermodynamics parameters
• T - air temperature
• p - air pressure
• w - vertical velocity
• q_tot - total water specific content
• q_liq - liquid water specific content
• q_ice - ice water specific content
• N_liq - liquid water number concentration
• N_ice - ice water number concentration

Returns the mass of activated aerosol particles per mode of the aerosol size distribution.

total_N_activated(ap, ad, aip, tps, T, p, w, q_tot, q_liq, q_ice, N_liq, N_ice)

• ap - a struct with aerosol activation parameters
• ad - aerosol distribution struct
• aip - a struct with air properties
• tps - a struct with thermodynamics parameters
• T - air temperature
• p - air pressure
• w - vertical velocity
• q_tot - total water specific content
• q_liq - liquid water specific content
• q_ice - ice water specific content
• N_liq - liquid water number concentration
• N_ice - ice water number concentration

Returns the total number of activated aerosol particles.

total_M_activated(ap, ad, aip, tps, T, p, w, q_tot, q_liq, q_ice, N_liq, N_ice)

• ap - a struct with aerosol activation parameters
• ad - aerosol distribution struct
• aip - a struct with air properties
• tps - a struct with thermodynamics parameters
• T - air temperature
• p - air pressure
• w - vertical velocity
• q_tot - total water specific content
• q_liq - liquid water specific content
• q_ice - ice water specific content
• N_liq - liquid water number concentration
• N_ice - ice water number concentration

Returns the total mass of activated aerosol particles.

Call artifacts from Artifacts.toml

AIDA_ice_nucleation(data_file_name)

• data_file_name - name of the data file on Caltech box.

Returns the filepath of the data file in Caltech box.

Parameterization for heterogenous cloud ice nucleation.

dust_activated_number_fraction(dust, ip, Si, T)

• dust - a struct with dust parameters
• ip - a struct with cloud ice nucleation parameters
• Si - ice saturation ratio
• T - air temperature [K]

Returns the number fraction of mineral dust particles acting as deposition nuclei (n ice nuclei / n dust particles). From Mohler et al 2006 Table 2 (averages from different measurements excluding those where a was not measured)

MohlerDepositionRate(dust, ip, S_i, T, dSi_dt, N_aer)

• dust - a struct with dust parameters
• ip - a struct with cloud ice nucleation parameters
• Si - ice saturation
• T - ambient temperature
• dSi_dt - change in ice saturation over time
• N_aer - number of unactivated aerosols

Returns the cloud ice nucleation rate from deposition. From Mohler et al 2006 equation 5.

deposition_J(dust, Δa_w)

• dust - a struct with dust parameters
• Δa_w - change in water activity [unitless].

Returns the deposition nucleation rate coefficient, J, in m^-2 s^-1 for different minerals in liquid droplets. The free parameters m and c are derived from China et al (2017) see DOI: 10.1002/2016JD025817 Returns zero for unsupported aerosol types.

ABIFM_J(dust, Δa_w)

• dust - a struct with dust parameters
• Δa_w - change in water activity [unitless].

Returns the immersion freezing nucleation rate coefficient, J, in m^-2 s^-1 for different minerals in liquid droplets. The free parameters m and c are taken from Knopf & Alpert 2013 see DOI: 10.1039/C3FD00035D Returns zero for unsupported aerosol types.

P3_deposition_N_i(ip, T)

• ip - a struct with ice nucleation parameters,
• T - air temperature [K].

Returns the number of ice crystals nucleated via deposition nucleation with units of m^-3. From Thompson et al 2004 eqn 2 as used in Morrison & Milbrandt 2015.

P3_het_N_i(ip, T, N_l, B, V_l, a, Δt)

• ip - a struct with ice nucleation parameters,
• T - air temperature [K],
• N_l - number of droplets [m^-3],
• B - water-type dependent parameter [cm^-3 s^-1],
• V_l - volume of droplets to be frozen [m^3],
• a - empirical parameter [C^-1],
• Δt - timestep.

Returns the number of ice nucleated within Δt via heterogeneous freezing with units of m^-3. From Pruppacher & Klett 1997 eqn (9-51) as used in Morrison & Milbrandt 2015.

INP_concentration_frequency(params,INPC,T)

• params - a struct with INPC(T) distribution parameters
• INPC - concentration of ice nucleating particles [m^-3]
• T - air temperature [K]

Returns the relative frequency of a given INP concentration, depending on the temperature. Based on Frostenberg et al., 2023. See DOI: 10.5194/acp-23-10883-2023

INP_concentration_mean(T)

• T - air temperature [K]

Returns the logarithm of mean INP concentration (in m^-3), depending on the temperature. Based on the function μ(T) in Frostenberg et al., 2023. See DOI: 10.5194/acp-23-10883-2023

Parameterization for homogeneous cloud ice nucleation

homogeneous_J_cubic(ip, Δa_w)

• ip - a struct with cloud ice nucleation parameters
• Δa_w - change in water activity [-].

Returns the homogeneous freezing nucleation rate coefficient, J, in m^-3 s^-1 for sulphuric acid solutions. Parameterization based on Koop 2000, DOI: 10.1038/35020537.

homogeneous_J_linear(ip, Δa_w)

• ip - a struct with ice nucleation parameters
• Δa_w - change in water activity [-].

Returns the homogeneous freezing nucleation rate coefficient, J, in m^-3 s^-1 for sulphuric acid solutions. Parameterization derived from a linear fit of the Koop 2000 parameterization, DOI: 10.1038/35020537.

Cloud diagnostics

• radar reflectivity (1-moment and 2-moment)
• effective radius (1-moment and 2-moment)

radar_reflectivity_1M(precip, q, ρ)

• precip - struct with 1-moment microphysics rain free parameters
• q - specific content of rain
• ρ - air density

Returns logarithmic radar reflectivity for the 1-moment microphysics based on the assumed rain particle size distribution. Normalized by the reflectivty of 1 millimiter drop in a volume of 1m3. The values are clipped at -150 dBZ.

radar_reflectivity_2M(structs, q_lcl, q_rai, N_lcl, N_rai, ρ_air)

- `structs` - structs microphysics 2-moment with SB2006 cloud droplets
              and raindrops size distributions parameters
- `q_lcl` - cloud liquid water specific content
- `q_rai` - rain water specific content
- `N_lcl` - cloud droplet number density
- `N_rai` - rain droplet number density
- `ρ_air` - air density

Returns logarithmic radar reflectivity for the 2-moment microphysics SB2006 based on the assumed cloud and rain particle size distribuions. Normalized by the reflectivty of 1 millimiter drop in a volume of 1m3. The values are clipped at -150 dBZ.

effective_radius_const(cloud_params)

• cloud_params - a struct with cloud liquid or cloud ice parameters

Returns a constant assumed effective radius for clouds

effective_radius_Liu_Hallet_97(q_lcl, q_rai, N_lcl, N_rai, ρ_air, ρ_w)

- `q_lcl` - cloud water specific content
- `q_rai` - rain water specific content
- `N_lcl` - cloud droplet number density
- `N_rai` - rain droplet number density
- `ρ_air` - air density
- `ρ_w` - water density

Returns effective radius using the "1/3" power law from Liu and Hallett (1997). If not provided by the user, it is assumed that there is no rain present and that the cloud droplet number concentration is 100 1/cm3.

effective_radius_2M(structs, q_lcl, q_rai, N_lcl, N_rai, ρ_air)

- `structs` - structs with SB2006 cloud droplets and raindrops
            size distribution parameters
- `q_lcl` - cloud liquid water specific content
- `q_rai` - rain water specific content
- `N_lcl` - cloud droplet number density
- `N_rai` - rain droplet number density
- `ρ_air` - air density

Returns effective radius for the 2-moment microphysics scheme. Computed based on the assumed cloud and rain particle size distributions.

Module for functions shared by different parameterizations.

G_func_liquid(air_props, tps, T)

• air_props - struct with air parameters
• tps - struct with thermodynamics parameters
• T - air temperature

Utility function combining thermal conductivity and vapor diffusivity effects for vapor to liquid phase change.

G_func_ice(air_props, tps, T)

• air_props - struct with air parameters
• tps - struct with thermodynamics parameters
• T - air temperature

Utility function combining thermal conductivity and vapor diffusivity effects. for vapor to ice phase change.

logistic_function(x, x_0, k)

• x - independent variable
• x_0 - threshold value for x
• k - growth rate of the curve, characterizing steepness of the transition

Returns the value of the logistic function for smooth transitioning at thresholds. This is a normalized curve changing from 0 to 1 while x varies from 0 to Inf (for positive k). For x < 0 the value at x = 0 (zero) is returned. For x_0 = 0 H(x) is returned.

logistic_function_integral(x, x_0, k)

• x - independent variable
• x_0 - threshold value for x
• k - growth rate of the logistic function, characterizing steepness of the transition

Returns the value of the indefinite integral of the logistic function, for smooth transitioning of piecewise linear profiles at thresholds. This curve smoothly transition from y = 0 for 0 < x < x0 to y = x - x0 for x_0 < x.

H2SO4_soln_saturation_vapor_pressure(prs, x, T)

• prs - a struct with H2SO4 solution free parameters
• x - wt percent sulphuric acid [unitless]
• T - air temperature [K].

Returns the saturation vapor pressure above a sulphuric acid solution droplet in Pa. x is, for example, 0.1 if droplets are 10 percent sulphuric acid by weight

a_w_xT(H2SO4_prs, tps, x, T)

• H2SO4_prs - a struct with H2SO4 solution free parameters
• tps - a struct with thermodynamics parameters
• x - wt percent sulphuric acid [unitless]
• T - air temperature [K].

Returns water activity of H2SO4 containing droplet. x is, for example, 0.1 if droplets are 10 percent sulphuric acid by weight.

a_w_eT(tps, e, T)

• tps - struct with thermodynamics parameters
• e - partial pressure of water [Pa]
• T - air temperature [K].

Returns water activity of pure water droplet. Valid when droplet is in equilibrium with surroundings.

a_w_ice(tps, T)

• tps - struct with thermodynamics parameters
• T - air temperature [K].

Returns water activity of ice.

Chen2022_vel_coeffs(coeffs, ρₐ)
Chen2022_vel_coeffs(coeffs, ρₐ, ρᵢ)

Compute the coefficients for the Chen 2022 terminal velocity parametrization.

Arguments

• coeffs: a struct with terminal velocity free parameters
  • CMP.Chen2022VelTypeRain: Fetch from Table B1
  • CMP.Chen2022VelTypeSmallIce: Fetch from Table B2
  • CMP.Chen2022VelTypeLargeIce: Fetch from Table B4
• ρₐ: air density [kg/m³]
• ρᵢ: apparent density of ice particles [kg/m³], only used for CMP.Chen2022VelTypeSmallIce and CMP.Chen2022VelTypeLargeIce

See [12] for more details.

Chen2022_monodisperse_pdf(a, b, c)

Arguments

• a, b, c: free parameters defined in [12]

Returns

• pdf(D): The monodisperse particle distribution function as a function of diameter, D, in [m/s].

Chen2022_exponential_pdf(a, b, c, λ_inv, k)

• a, b, c, - free parameters defined in Chen etl al 2022
• λ_inv - inverse of the size distribution parameter
• k - size distribution moment for which we compute the bulk fall speed

Returns the addends of the bulk fall speed of rain or ice particles following Chen et al 2022 DOI: 10.1016/j.atmosres.2022.106171 in [m/s]. Assuming exponential size distribution and hence μ=0.

volume_sphere_D(D)

Calculate the volume of a sphere with diameter D.

\[V = D^3 * π / 6\]

See also volume_sphere_R.

volume_sphere_R(R)

Calculate the volume of a sphere with radius R.

\[V = (2R)^3 * π / 6\]

See also volume_sphere_D.

ventilation_factor(vent, aps, v_term)

Returns a function that computes the ventilation factor for a particle as a function of its diameter, D.

The ventilation factor parameterizes the increase in the mass and heat exchange for falling particles.

Arguments

• vent: Ventilation parameterization constants, CMP.VentilationFactor
• aps: Parameters with air properties, CMP.AirProperties
• v_term: A function v_term(D) that returns the terminal velocity of a particle with diameter D

Returns

• F_v(D): The ventilation factor as a function of diameter, D

See e.g. [10] Eq. (24), or CMP.VentilationFactor, for the definition of the ventilation factor.

Parameters

A module for CloudMicrophysics.jl free parameters.

ParametersType{FT}

The top-level super-type for all cloud microphysics parameters

AerosolType{FT}

The top-level super-type for all aerosol properties

AerosolDistributionType

The top-level super-type for all aerosol distribution types

CloudCondensateType{FT}

The top-level super-type for cloud condensate types (liquid and ice)

PrecipitationType{FT}

The top-level super-type for precipitation types (rain and snow)

TerminalVelocityType{FT}

The top-level super-type for terminal velocity parameterizations

Precipitation2MType

The top-level super-type for 2-moment precipitation parameterizations

AirProperties{FT}

Parameters with air properties.

Fields

• K_therm: thermal conductivity of air [w/m/K]

• D_vapor: diffusivity of water vapor [m2/s]

• ν_air: kinematic viscosity of air [m2/s]

WaterProperties{FT}

Parameters with water properties.

Fields

• ρw: density of liquid water [kg/m3]

• ρi: density of ice [kg/m3]

ArizonaTestDust{FT}

Parameters for Arizona Test Dust from Mohler et al, 2006. DOI: 10.5194/acp-6-3007-2006

Fields

• S₀_warm: S₀ for T > T_thr [-]

• S₀_cold: S₀ for T < T_thr [-]

• a_warm: a for T > T_thr [-]

• a_cold: a for T < T_thr [-]

• deposition_m: m coefficient for deposition nucleation J [-]

• deposition_c: c coefficient for deposition nucleation J [-]

• ABIFM_m: m coefficient for immersion freezing J [-]

• ABIFM_c: c coefficient for immersion freezing J [-]

DesertDust{FT}

Parameters for desert dust from Knopf and Alpert 2013 DOI: 10.1039/C3FD00035D and from Mohler et al, 2006 DOI: 10.5194/acp-6-3007-2006

Fields

• S₀_warm: S₀ for T > T_thr [-]

• S₀_cold: S₀ for T < T_thr [-]

• a_warm: a for T > T_thr [-]

• a_cold: a for T < T_thr [-]

• ABIFM_m: m coefficient for immersion freezing J [-]

• ABIFM_c: c coefficient for immersion freezing J [-]

AsianDust{FT}

Parameters for Asian Dust

Fields

• deposition_m: m coefficient for deposition nucleation J [-]

• deposition_c: c coefficient for deposition nucleation J [-]

• ABIFM_m: m coefficient for immersion freezing J [-]

• ABIFM_c: c coefficient for immersion freezing J [-]

MiddleEasternDust{FT}

Parameters for Middle Eastern Dust

Fields

• ABIFM_m: m coefficient for immersion freezing J [-]

• ABIFM_c: c coefficient for immersion freezing J [-]

SaharanDust{FT}

Parameters for Saharan Dust

Fields

• deposition_m: m coefficient for deposition nucleation J [-]

• deposition_c: c coefficient for deposition nucleation J [-]

Illite{FT}

Parameters for illite from Knopf and Alpert 2013 DOI: 10.1039/C3FD00035D

Fields

• deposition_m: m coefficient for deposition nucleation J [-]

• deposition_c: c coefficient for deposition nucleation J [-]

• ABIFM_m: m coefficient for immersion freezing J [-]

• ABIFM_c: c coefficient for immersion freezing J [-]

Kaolinite{FT}

Parameters for kaolinite from Knopf and Alpert 2013 DOI: 10.1039/C3FD00035D and China et al 2017 DOI: 10.1002/2016JD025817

Fields

• deposition_m: m coefficient for deposition nucleation J [-]

• deposition_c: c coefficient for deposition nucleation J [-]

• ABIFM_m: m coefficient for immersion freezing J [-]

• ABIFM_c: c coefficient for immersion freezing J [-]

Feldspar{FT}

Parameters for Feldspar from Alpert et al 2022 DOI: 10.1039/D1EA00077B

Fields

• deposition_m: m coefficient for deposition nucleation J [-]

• deposition_c: c coefficient for deposition nucleation J [-]

Ferrihydrite{FT}

Parameters for Ferrihydrite from Alpert et al 2022 DOI: 10.1039/D1EA00077B

Fields

• deposition_m: m coefficient for deposition nucleation J [-]

• deposition_c: c coefficient for deposition nucleation J [-]

Dust{FT}

Parameters for generic dust.

Fields

• deposition_m: m coefficient for deposition nucleation J [-]

• deposition_c: c coefficient for deposition nucleation J [-]

• ABIFM_m: m coefficient for immersion freezing J [-]

• ABIFM_c: c coefficient for immersion freezing J [-]

Seasalt{FT}

Parameters for seasalt

Fields

• M: molar mass [kg/mol]

• ρ: density [kg/m3]

• ϕ: osmotic coefficient [-]

• ν: ion number [-]

• ϵ: water soluble mass fraction [-]

• κ: hygroscopicity parameter [-]

Sulfate{FT}

Parameters for sulfate aerosol

Fields

• M: molar mass [kg/mol]

• ρ: density [kg/m3]

• ϕ: osmotic coefficient [-]

• ν: ion number [-]

• ϵ: water soluble mass fraction [-]

• κ: hygroscopicity parameter [-]

AerosolActivationParameters{FT}

Parameters for Abdul-Razzak and Ghan 2000 aerosol activation scheme DOI: 10.1029/1999JD901161

Fields

• M_w: molar mass of water [kg/mol]

• R: gas constant [J/mol/K]

• ρ_w: cloud water density [kg/m3]

• ρ_i: cloud ice density [kg/m3]

• σ: surface tension of water [N/m]

• g: gravitational_acceleration [m/s2]

• f1: scaling coefficient in Abdul-Razzak and Ghan 2000 [-]

• f2: scaling coefficient in Abdul-Razzak and Ghan 2000 [-]

• g1: scaling coefficient in Abdul-Razzak and Ghan 2000 [-]

• g2: scaling coefficient in Abdul-Razzak and Ghan 2000 [-]

• p1: power of (zeta / eta) in Abdul-Razzak and Ghan 2000 [-]

• p2: power of (S_m^2 / (zeta + 3 * eta)) in Abdul-Razzak and Ghan 2000 [-]

IceNucleationParameters{FT, DEP, HOM, P3_type}

Parameters for ice nucleation

Fields

• deposition

• homogeneous

• p3

Frostenberg2023{FT}

Parameters for frequency distribution of INP concentration DOI: 10.5194/acp-23-10883-2023

Fields

• σ: standard deviation

• a: coefficient

• b: coefficient

H2SO4SolutionParameters{FT}

Parameters for water activity of H2SO4 solutions from Luo et al 1995. DOI: 10.1029/94GL02988

Fields

• T_max: max temperature for which the parameterization is valid [K]

• T_min: min temperature for which the parameterization is valid [K]

• w_2: coefficient [-]

• c1: coefficient [-]

• c2: coefficient [-]

• c3: coefficient [-]

• c4: coefficient [-]

• c5: coefficient [-]

• c6: coefficient [-]

• c7: coefficient [-]

Mohler2006{FT}

Parameters for ice nucleation from Mohler et al 2006 DOI: 10.5194/acp-6-3007-2006

Fields

• Sᵢ_max: max allowed supersaturation [-]

• T_thr: threshold temperature [K]

Koop2000{FT}

Parameters for ice nucleation from Koop et al 2000 DOI: 10.1038/35020537

Fields

• Δa_w_min: min Δaw [-]

• Δa_w_max: max Δaw [-]

• c₁: coefficient [-]

• c₂: coefficient [-]

• c₃: coefficient [-]

• c₄: coefficient [-]

• linear_c₁: coefficient [-]

• linear_c₂: coefficient [-]

H2S04NucleationParameters{FT}

Parameters for pure sulfuric acid nucleation from Dunne et al 1016 DOI:10.1126/science.aaf2649

Fields

• p_b_n

• p_b_i

• u_b_n

• u_b_i

• v_b_n

• v_b_i

• w_b_n

• w_b_i

• p_t_n

• p_t_i

• u_t_n

• u_t_i

• v_t_n

• v_t_i

• w_t_n

• w_t_i

• p_A_n

• p_A_i

• a_n

• a_i

OrganicNucleationParameters{FT}

Parameters for pure organic nucleation from Kirkby 2016 DOI: 10.1038/nature17953

Fields

• a_1

• a_2

• a_3

• a_4

• a_5

• Y_MTO3

• Y_MTOH

• k_MTO3

• k_MTOH

• exp_MTO3

• exp_MTOH

MixedNucleationParameters{FT}

Parameters for mixed organic and sulfuric acid nucleation from Riccobono et al 2014 DOI:10.1126/science.1243527

Fields

• k_H2SO4org

• k_MTOH

• exp_MTOH

Parameters0M{FT}

Parameters for zero-moment bulk microphysics scheme

Fields

• τ_precip: precipitation timescale [s]

• qc_0: condensate specific content precipitation threshold [-]

• S_0: supersaturation precipitation threshold [-]

ParticlePDFSnow{FT}

A struct with snow size distribution parameters

Fields

• μ: snow size distribution coefficient [1/m4]

• ν: snow size distribution coefficient [-]

ParticlePDFIceRain{FT}

A struct with snow size distribution parameters

Fields

• n0: Size distribution coefficient [1/m4]

ParticleMass{FT}

A struct with coefficients of the assumed mass(size) relationship for particles

m(r) = m0 χm (r/r0)^(me + Δm)

Fields

• r0: particle length scale [m]

• m0: mass size relation coefficient [kg]

• me: mass size relation coefficient [-]

• Δm: mass size relation coefficient [-]

• χm: mass size relation coefficient [-]

ParticleArea{FT}

A struct with coefficients of the assumed crosssectionarea(size) relationship for particles

a(r) = a0 χa (r/r0)^(ae + Δa)

Fields

• a0: cross section size relation coefficient [m2]

• ae: cross section size relation coefficient [-]

• Δa: cross section size relation coefficient [-]

• χa: cross section size relation coefficient [-]

SnowAspectRatio{FT}

A struct with aspect ratio coefficients

Fields

• ϕ: aspect ratio [-]

• κ: power law coefficient in terminal velocity parameterization from Chen et al 2022 [-]

Ventilation{FT}

A struct with ventilation coefficients

Fields

• a: ventilation coefficient a [-]

• b: ventilation coefficient b [-]

Acnv1M{FT}

A struct with autoconversion parameters

Fields

• τ: autoconversion timescale [s]

• q_threshold: condensate specific content autoconversion threshold [-]

• k: threshold smooth transition steepness [-]

CloudLiquid{FT}

The parameters and type for cloud liquid water condensate

Fields

• τ_relax: condensation evaporation non_equil microphysics relaxation timescale [s]

• ρw: water density [kg/m3]

• r_eff: effective radius [m]

CloudIce{FT, MS}

The parameters and type for cloud ice condensate

Fields

• pdf: a struct with size distribution parameters

• mass: a struct with mass size relation parameters

• r0: particle length scale [m]

• r_ice_snow: ice snow threshold radius [m]

• τ_relax: deposition sublimation non_equil microphysics relaxation timescale [s]

• ρᵢ: cloud ice apparent density [kg/m3]

• r_eff: effective radius [m]

Rain{FT, MS, AR, VT, AC}

The parameters and type for rain

Fields

• pdf: a struct with size distribution parameters

• mass: a struct with mass size relation parameters

• area: a struct with cross section size relation parameers

• vent: a struct with ventilation coefficients

• acnv1M: a struct with cloud water to rain autoconversion parameters

• r0: particle length scale [m]

Snow{FT, PD, MS, AR, VT, AP, AC}

The parameters and type for snow

Fields

• pdf: a struct with size distribution parameters

• mass: a struct with mass size relation parameters

• area: a struct with cross section size relation parameers

• vent: a struct with ventilation coefficients

• aspr: a struct with aspect ratio parameters

• acnv1M: a struct with ice to snow autoconversion parameters

• r0: particle length scale [m]

• T_freeze: freezing temperature of water [K]

• ρᵢ: snow apparent density [kg/m3]

CollisionEff{FT}

Collision efficiency parameters for the 1-moment scheme

Fields

• e_lcl_rai: cloud liquid-rain collision efficiency [-]

• e_lcl_sno: cloud liquid-snow collision efficiency [-]

• e_icl_rai: cloud ice-rain collision efficiency [-]

• e_icl_sno: cloud ice-snow collision efficiency [-]

• e_rai_sno: rain-snow collision efficiency [-]

KK2000

The type and parameters for 2-moment precipitation formation by Khairoutdinov and Kogan (2000)

DOI:10.1175/1520-0493(2000)128<0229:ANCPPI>2.0.CO;2

Fields

• acnv: Autoconversion parameters

• accr: Accretion parameters

AcnvKK2000

Khairoutdinov and Kogan (2000) autoconversion parameters

Fields

• A: Autoconversion coefficient A

• a: Autoconversion coefficient a

• b: Autoconversion coefficient b

• c: Autoconversion coefficient c

AccrKK2000

Khairoutdinov and Kogan (2000) accretion parameters

Fields

• A: Accretion coefficient A

• a: Accretion coefficient a

• b: Accretion coefficient b

B1994

The type and parameter for 2-moment precipitation formation by Beheng (1994) DOI: 10.1016/0169-8095(94)90020-5

Fields

• acnv: Autoconversion coeff C

• accr: Autoconversion coeff a

AcnvB1994

Beheng (1994) autoconversion parameters

Fields

• C: Autoconversion coeff C

• a: Autoconversion coeff a

• b: Autoconversion coeff b

• c: Autoconversion coeff c

• N_0: Autoconversion coeff N_0

• d_low: Autoconversion coeff d_low

• d_high: Autoconversion coeff d_high

• k: Threshold for smooth tranistion steepness

AccrB1994

Beheng (1994) accretion parameters

Fields

• A: Accretion coefficient A

TC1980

The type and parameters for 2-moment precipitation formation by Tripoli and Cotton (1980)

DOI: 10.1175/1520-0450(1980)019<1037:ANIOSF>2.0.CO;2

Fields

• acnv: Autoconversion parameters

• accr: Accretion parameters

AcnvTC1980

Tripoli and Cotton (1980) autoconversion parameters

Fields

• a: Autoconversion coefficient a

• b: Autoconversion coefficient b

• D: Autoconversion coefficient D

• r_0: Autoconversion coefficient r_0

• me_liq: Autoconversion coefficient me_liq

• m0_liq_coeff: Autoconversion coefficient m0liqcoeff

• k: Threshold for smooth tranistion steepness

AccrTC1980

Tripoli and Cotton (1980) accretion parameters

Fields

• A: Accretion coefficient A

LD2004

The type and parameters for 2-moment precipitation formation by Liu and Daum (2004)

DOI: 10.1175/1520-0469(2004)061<1539:POTAPI>2.0.CO;2

Fields

• R_6C_0: Autoconversion coefficient R6C0

• E_0: Autoconversion coefficient E_0

• ρ_w: liquid water density [kg/m3]

• k: Threshold for smooth tranistion steepness

VarTimescaleAcnv

The type for 2-moment precipitation formation based on the 1-moment parameterization with variable time scale Azimi et al (2023)

Fields

• τ: Timescale [s]

• α: Powerlaw coefficient [-]

SB2006

The type and parameters for 2-moment precipitation formation by Seifert and Beheng (2006). The pdf_r type choses between running with or without limiters on raindrop size distribution parameters

DOI: 10.1007/s00703-005-0112-4

Fields

• pdf_c: Cloud particle size distribution parameters

• pdf_r: Rain particle size distribution parameters

• acnv: Autoconversion parameters

• accr: Accretion parameters

• self: Rain selfcollection parameters

• brek: Rain breakup parameters

• evap: Rain evaporation parameters

• numadj: Number concentration adjustment parameter

RainParticlePDF_SB2006

Abstract type for the size distribution parameters of rain particles

See RainParticlePDF_SB2006_limited and RainParticlePDF_SB2006_notlimited for the concrete types.

RainParticlePDF_SB2006_limited

Rain size distribution parameters from SB2006 including the limiters on drop maximum mass and the size distribution coefficinets N0 and lambda

Fields

• νr: Raindrop size distribution coefficient νr

• μr: Raindrop size distribution coefficient μr

• xr_min: Raindrop minimal mass

• xr_max: Raindrop maximum mass

• N0_min: Raindrop size distribution coefficient N0 min

• N0_max: Raindrop size distribution coefficient N0 max

• λ_min: Raindrop size distribution coefficient lambda min

• λ_max: Raindrop size distribution coefficient lambda max

• ρw: Cloud liquid water density [kg/m3]

• ρ0: Reference air density [kg/m3]

RainParticlePDF_SB2006

Rain size distribution parameters from SB2006 but without the limiters

Fields

• νr: Raindrop size distribution coefficient νr

• μr: Raindrop size distribution coefficient μr

• xr_min: Raindrop minimum mass

• xr_max: Raindrop maximum mass

• ρw: Cloud liquid water density [kg/m3]

• ρ0: Reference air density [kg/m3]

CloudParticlePDF_SB2006

Cloud droplets size distribution parameters from SB2006

Fields

• νc: Cloud droplet size distribution coefficient νc

• μc: Cloud droplet size distribution coefficient μc

• xc_min: Cloud droplets minimum mass

• xc_max: Cloud droplets maximum mass

• ρw: Cloud liquid water density [kg/m3]

AcnvSB2006

Autoconversion parameters from SB2006

Fields

• kcc: Collection kernel coefficient

• x_star: Minimum mass of rain droplets

• ρ0: Reference air density [kg/m3]

• A: Autoconversion correcting function coeff A

• a: Autoconversion correcting function coeff a

• b: Autoconversion correcting function coeff b

AccrSB2006

Accretion parameters from SB2006

Fields

• kcr: Collection kernel coefficient Kcr

• τ0: Accretion correcting function coefficient τ_0

• ρ0: Reference air density [kg/m3]

• c: Accretion correcting function coefficient c

SelfColSB2006

Rain selfcollection parameters from SB2006

Fields

• krr: Collection kernel coefficient krr

• κrr: Collection kernel coefficient kappa rr

• d: Raindrop self collection coefficient d

BreakupSB2006

Rain breakup parameters from SB2006

Fields

• Deq: Raindrop equilibrium mean diamater

• Dr_th: Raindrop breakup mean diamater threshold

• kbr: Raindrops breakup coefficient kbr

• κbr: Raindrops breakup coefficient kappa br

EvaporationSB2006

Rain evaporation parameters from SB2006

Fields

• av: Ventillation coefficient a

• bv: Ventillation coefficient b

• α: Rain evapoartion coefficient α

• β: Rain evapoartion coefficient β

• ρ0: Reference air density [kg/m3]

NumberAdjustmentHorn2012

Number concentration adjustment parameter from Horn (2012, DOI: 10.5194/gmd-5-345-2012)

Fields

• τ: Number concentration adjustment timescale [s]

Blk1MVelType

The type for precipitation terminal velocity from the simple 1-moment scheme (defined for rain and snow)

Fields

• rain

• snow

Blk1MVelTypeRain

The type for precipitation terminal velocity from the simple 1-moment scheme for rain

Fields

• r0: particle length scale [m]

• ve: rain terminal velocity size relation coefficient [-]

• Δv: rain terminal velocity size relation coefficient [-]

• χv: rain terminal velocity size relation coefficient [-]

• ρw: cloud water density [kg/m3]

• C_drag: rain drop drag coefficient [-]

• grav: gravitational acceleration [m/s2]

Blk1MVelTypeSnow

The type for precipitation terminal velocity from the simple 1-moment scheme for snow

Fields

• r0: particle length scale [m]

• ve: snow terminal velocity size relation coefficient [-]

• Δv: snow terminal velocity size relation coefficient [-]

• χv: snow terminal velocity size relation coefficient [-]

• v0: snow terminal velocity size relation coefficient [m/s]

StokesRegimeVelType

The type for precipitation terminal velocity in the Stokes regime (Re < 1)

Fields

• ρw

• ν_air

• grav

SB2006VelType

The type for precipitation terminal velocity from Seifert and Beheng 2006

Fields

• ρ0

• aR

• bR

• cR

• ρw

• ν_air

• grav

Chen2022VelType

The type for precipitation terminal velocity from Chen et al 2022 DOI: 10.1016/j.atmosres.2022.106171 (defined for rain, snow and cloud ice)

Fields

• rain

• small_ice

• large_ice

Chen2022VelTypeSmallIce

The type for precipitation terminal velocity from Chen et al 2022 for small ice. See Table B3 for parameter definitions. DOI: 10.1016/j.atmosres.2022.106171

Fields

• A

• B

• C

• E

• F

• G

• cutoff: cutoff for small vs large ice particle dimension [m]

Chen2022VelTypeLargeIce

The type for precipitation terminal velocity from Chen et al 2022 for large ice. See Table B4 for parameter definitions. DOI: 10.1016/j.atmosres.2022.106171

Fields

• A

• B

• C

• E

• F

• G

• H

• cutoff: cutoff for small vs large ice particle dimension [m]

Chen2022VelTypeRain

The type for precipitation terminal velocity from Chen et al 2022 for rain. See Table B1 for parameter definitions. DOI: 10.1016/j.atmosres.2022.106171

Fields

• ρ0

• a

• a3_pow

• b

• b_ρ

• c

precipitation_susceptibility_autoconversion(param_set, scheme, q_lcl, q_rai, ρ, N_lcl)

• scheme - type for 2-moment rain autoconversion parameterization
• q_lcl - cloud liquid water specific content
• q_rai - rain water specific content
• ρ - air density
• N_lcl - cloud droplet number density

Returns the precipitation susceptibility rates due to autoconversion as a precip_susceptibility_rates object, using automatic differentiation. Works for any 2-moment scheme, as long as autoconversion is defined for it.

precipitation_susceptibility_accretion(param_set, scheme, q_lcl, q_rai, ρ, N_lcl)

• scheme - type for 2-moment rain autoconversion parameterization
• q_lcl - cloud liquid water specific content
• q_rai - rain water specific content
• ρ - air density
• N_lcl - cloud droplet number density

Returns the precipitation susceptibility rates due to accretion as a precip_susceptibility_rates object, using automatic differentiation. Works for any 2-moment scheme, as long as accretion is defined for it.