Microphysics

The Microphysics.jl module describes a 1-moment bulk parameterization of cloud microphysical processes. The module is based on the ideas of Kessler_1995, Grabowski_1998 and Kaul_et_al_2015.

The cloud microphysics variables are expressed as specific humidities:

• q_tot - total water specific humidity,
• q_vap - water vapor specific humidity,
• q_liq - cloud water specific humidity,
• q_ice - cloud ice specific humidity,
• q_rai - rain specific humidity,
• q_sno - snow specific humidity.

Assumed particle size relationships

Particles are assumed to follow mass(radius), cross section(radius), and terminal velocity(radius) relationships defined as power laws. The coefficients are defined in the CLIMAParameters.jl package and are shown in the table below. For rain and ice they correspond to spherical liquid water drops and ice particles, respectively. There is no assumed particle shape for snow, and the relationships are empirical.

where:

• $r$ is the particle radius,
• $r_0$ is the typical particle radius used to nondimensionalize,
• $m_0, \, m_e, \, a_0, \, a_e, \, v_0, \, v_e \,$ are the default coefficients,
• $\chi_m$, $\Delta_m$, $\chi_a$, $\Delta_a$, $\chi_v$, $\Delta_v$ are the coefficients that can be used during model calibration to expand around the default values. Without calibration all $\chi$ parameters are set to 1 and all $\Delta$ parameters are set to 0.

The above coefficients, similarly to all other microphysics parameters, are not hardcoded in the final microphysics parameterizations. The goal is to allow easy flexibility when calibrating the scheme. With that said, the assumption about the shape of the particles is used three times when deriving the microphysics formulae:

• The mass transfer equation (\ref{eq:mass_rate}) used in snow autoconversion, rain evaporation, snow sublimation and snow melt rates is derived assuming spherical particle shapes. To correct for non-spherical shape it should be multiplied by a function of the particle aspect ratio.
• The geometric collision kernel used for deriving rain-snow accretion rate assumes that both colliding particles are spherical. It does not take into account the reduced cross-section of snow particles that is used when modeling snow - cloud liquid water and snow - cloud ice accretion.
• In the definition of the Reynolds number that is used when computing ventilation factors.

where:

• $\rho_{water}$ is the density of water,
• $\rho_{ice}$ is the density of ice.

The terminal velocity of an individual rain drop is defined by the balance between the gravitational acceleration (taking into account the density difference between water and air) and the drag force. Therefore the $v_0^{rai}$ is defined as

where:

• $g$ is the gravitational acceleration,
• $C_{drag}$ is the drag coefficient,
• $\rho$ is the density of air.

Assumed particle size distributions

The particle size distributions are assumed to follow Marshall-Palmer distribution Marshall_and_Palmer_1948 eq. 1:

where:

• $n_{0}$ and $\lambda$ are the Marshall-Palmer distribution parameters.

The $n_0$ for rain and ice is assumed constant. The $n_0$ for snow is defined as

where:

• $\mu^{sno}$ and $\nu^{sno}$ are the coefficients
• $\rho_0$ is the typical air density used to nondimensionalize the equation and is equal to $1 \, kg/m^3$

The coefficients are defined in CLIMAParameters.jl package and are shown in the table below.

The $\lambda$ parameter is defined as

where:

• $q$ is rain, snow or ice specific humidity
• $\chi_m$, $m_0$, $m_e$, $\Delta_m$, $r_0$, and $n_0$ are the corresponding mass(radius) and size distribution parameters
• $\Gamma()$ is the gamma function

The cloud-ice size distribution is used when computing snow autoconversion rate and rain sink due to accretion. In other derivations cloud ice, similar to cloud liquid water, is treated as continuous.

Parameterized processes

Parameterized processes include:

• diffusion of water vapour on cloud droplets and cloud ice crystals modeled as a relaxation to equilibrium,
• autoconversion of rain and snow,
• accretion,
• evaporation of rain water,
• sublimation, vapor deposition and melting of snow,
• sedimentation of rain and snow with mass weighted average terminal velocity (cloud water and cloud ice are part of the working fluid and do not sediment).

Parameters used in the parameterization are defined in CLIMAParameters.jl package. They consist of:

Ventilation factor

The ventilation factor parameterizes the increase in the mass and heat exchange for falling particles. Following Seifert_and_Beheng_2006 eq. 24 the ventilation factor is defined as:

where:

• $a_{vent}$, $b_{vent}$ are coefficients,
• $N_{Sc}$ is the Schmidt number,
• $N_{Re}$ is the Reynolds number of a falling particle.

The Schmidt number is assumed constant:

where:

• $\nu_{air}$ is the kinematic viscosity of air,
• $D_{vapor}$ is the diffusivity of water.

The Reynolds number of a spherical drop is defined as:

Applying the terminal velocity(radius) relationship results in

Terminal velocity

The mass weighted terminal velocity $v_t$ is defined following Ogura_and_Takahashi_1971:

Integrating over the assumed Marshall-Palmer distribution and using the $m(r)$ and $v_{term}(r)$ relationships results in

Cloud water condensation/evaporation

Condensation and evaporation of cloud liquid water is parameterized as a relaxation to equilibrium value at the current time step.

where:

• $q^{eq}_{liq}$ - liquid water specific humidity in equilibrium,
• $q_{liq}$ - liquid water specific humidity,
• $\tau_{cond\_evap}$ - relaxation timescale.

Cloud ice deposition/sublimation

Deposition and sublimation of cloud ice is parameterized as a relaxation to equilibrium value at the current time step.

where:

• $q^{eq}_{ice}$ - ice specific humidity in equilibrium,
• $q_{ice}$ - ice specific humidity,
• $\tau_{dep\_sub}$ - relaxation timescale.

Rain autoconversion

Rain autoconversion defines the rate of conversion form cloud liquid water to rain water due to collisions between cloud droplets. It is parameterized following Kessler_1995:

where:

• $q_{liq}$ - liquid water specific humidity,
• $\tau_{acnv}$ - timescale,
• $q_{liq\_threshold}$ - autoconversion threshold.

Snow autoconversion

Snow autoconversion defines the rate of conversion form cloud ice to snow due the growth of cloud ice by water vapor deposition. It is defined as the change of mass of cloud ice for cloud ice particles larger than threshold $r_{is}$. See Harrington_1995 for derivation.

The $\frac{dm}{dt}$ is obtained by solving the water vapor diffusion equation in spherical coordinates and linking the changes in temperature at the drop surface to the changes in saturated vapor pressure via the Clausius-Clapeyron equation, following Mason_1971.

For the simplest case of spherical particles and not taking into account ventilation effects:

where:

• $S(q_{vap}, q_{vap}^{sat}) = \frac{q_{vap}}{q_{vap}^{sat}}$ is saturation,
• $q_{vap}^{sat}$ is the saturation vapor specific humidity (over ice in this case),
• $G(T) = \left(\frac{L_s}{KT} \left(\frac{L_s}{R_v T} - 1 \right) + \frac{R_v T}{p_{vap}^{sat} D} \right)^{-1}$ combines the effects of thermal conductivity and water diffusivity.
• $L_s$ is the latent heat of sublimation,
• $K_{thermo}$ is the thermal conductivity of air,
• $R_v$ is the gas constant of water vapor,
• $D_{vapor}$ is the diffusivity of water vapor

Using eq. (\ref{eq:mass_rate}) and the assumed $m(r)$ relationship we obtain

Finally the snow autoconversion rate is computed as

Accretion

Accretion defines the rates of conversion between different categories due to collisions between particles.

For the case of collisions between cloud water (liquid water or ice) and precipitation (rain or snow) the sink of cloud water is defined as:

where:

• $c$ subscript indicates cloud water category (cloud liquid water or ice)
• $p$ subscript indicates precipitation category (rain or snow)
• $E_{cp}$ is the collision efficiency.

Integrating over the distribution yields:

where:

• $\Pi_{a, v}^p = a_0^p \, v_0^p \, \chi_a^p \, \chi_v^p$
• $\Sigma_{a, v}^p = a_e^p + v_e^p + \Delta_a^p + \Delta_v^p$

For the case of cloud liquid water and rain and cloud ice and snow collisions, the sink of cloud water becomes simply the source for precipitation. For the case of cloud liquid water and snow collisions for temperatures below freezing, the sink of cloud liquid water is a source for snow. For temperatures above freezing, the accreted cloud droplets along with some melted snow are converted to rain. In this case eq. (\ref{eq:accrfin}) describes the sink of cloud liquid water. The sink of snow is proportional to the sink of cloud liquid water with the coefficient $\frac{c_w}{L_f}(T - T_{freeze})$.

The collisions between cloud ice and rain create snow. The source of snow in this case is a sum of sinks from cloud ice and rain. The sink of cloud ice is defined by eq. (\ref{eq:accrfin}). The sink of rain is defined as:

where:

• $E_{ir}$ is the collision efficiency between rain and cloud ice
• $n_i$ and $n_r$ are the cloud ice and rain size distributions
• $m_r$, $a_r$ and $v_{term}$ are the mass(radius), cross section(radius) and terminal velocity(radius) relations for rain
• $r_i$ and $r_r$ mark integration over cloud ice and rain size distributions

Integrating eq.(\ref{eq:accr_ir}) yields:

where:

• $\Pi_{m, a, v}^{rai} = m_0^{rai} \, a_0^{rai} \, v_0^{rai} \, \chi_m^{rai} \, \chi_a^{rai} \, \chi_v^{rai}$
• $\Sigma_{m, a, v}^{rai} = m_e^{rai} + a_e^{rai} + v_e^{rai} + \Delta_m^{rai} + \Delta_a^{rai} + \Delta_v^{rai}$

Collisions between rain and snow result in snow in temperatures below freezing and in rain in temperatures above freezing. The source term is defined as:

where

• $i$ stands for rain ($T>T_{freezing}$) or snow ($T<T_{freezing}$)
• $j$ stands for snow ($T>T_{freezing}$) or rain ($T<T_{freezing}$)
• $a(r_i, r_j)$ is the crossection of the two colliding particles

There are two additional assumptions that we make to integrate eq.(\ref{eq:accr_sr1}):

• $\left|v_{term}(r_i) - v_{term}(r_j)\right| \approx \left| v_{ti} - v_{tj} \right|$ We approximate the terminal velocity difference for each particle pair with a difference between mass-weighted mean terminal velocities and move it outside of the integral. See the discussion in Ikawa_and_Saito_1991 page 88.

• We assume that $a(r_i, r_j) = \pi (r_i + r_j)^2$. This corresponds to a geometric formulation of the collision kernel, aka cylindrical formulation, see Wang_et_al_2005 for discussion.

The eq.(\ref{eq:accr_sr1}) can then be integrated as:

Rain evaporation and snow sublimation

We start from a similar equation as when computing snow autoconversion rate but integrate it from $0$ to $\infty$.

In contrast to eq.(\ref{eq:mass_rate}), now we are taking into account ventilation effects:

where:

• $F(r)$ is the rain drop ventilation factor defined in (\ref{eq:ventil_factor})
• saturation S is computed over water or ice

The final integral is:

For the case of rain we only consider evaporation ($S - 1 < 0$). For the case of snow we consider both the source term due to vapor deposition on snow ($S - 1 > 0$) and the sink due to vapor sublimation ($S - 1 < 0$).

Snow melt

If snow occurs in temperatures above freezing it is melting into rain. The sink for snow is parameterized again as

For snow melt

where:

• $F(r)$ is the ventilation factor defined in (\ref{eq:ventil_factor})
• $L_f$ is the latent heat of freezing.

If $T > T_{freeze}$:

Example figures

using Plots

using ClimateMachine.Microphysics
using ClimateMachine.Thermodynamics

using CLIMAParameters
using CLIMAParameters.Planet: R_d, planet_radius, grav, MSLP, molmass_ratio
using CLIMAParameters.Atmos.Microphysics
struct EarthParameterSet <: AbstractEarthParameterSet end
struct LiquidParameterSet <: AbstractLiquidParameterSet end
struct RainParameterSet <: AbstractRainParameterSet end
struct IceParameterSet <: AbstractIceParameterSet end
struct SnowParameterSet <: AbstractSnowParameterSet end

const param_set = EarthParameterSet()
const liquid_param_set = LiquidParameterSet()
const rain_param_set = RainParameterSet()
const ice_param_set = IceParameterSet()
const snow_param_set = SnowParameterSet()

# eq. 5d in Smolarkiewicz and Grabowski 1996
# https://doi.org/10.1175/1520-0493(1996)124<0487:TTLSLM>2.0.CO;2
function terminal_velocity_empirical(q_rai::DT, q_tot::DT, ρ::DT, ρ_air_ground::DT) where {DT<:Real}
    rr  = q_rai / (DT(1) - q_tot)
    vel = DT(14.34) * ρ_air_ground^DT(0.5) * ρ^-DT(0.3654) * rr^DT(0.1346)
    return vel
end

# eq. 5b in Smolarkiewicz and Grabowski 1996
# https://doi.org/10.1175/1520-0493(1996)124<0487:TTLSLM>2.0.CO;2
function accretion_empirical(q_rai::DT, q_liq::DT, q_tot::DT) where {DT<:Real}
    rr  = q_rai / (DT(1) - q_tot)
    rl  = q_liq / (DT(1) - q_tot)
    return DT(2.2) * rl * rr^DT(7/8)
end

# eq. 5c in Smolarkiewicz and Grabowski 1996
# https://doi.org/10.1175/1520-0493(1996)124<0487:TTLSLM>2.0.CO;2
function rain_evap_empirical(q_rai::DT, q::PhasePartition, T::DT, p::DT, ρ::DT) where {DT<:Real}

    ts_neq = TemperatureSHumNonEquil(param_set, T, ρ, q)
    q_sat  = q_vap_saturation(ts_neq)
    q_vap  = q.tot - q.liq
    rr     = q_rai / (DT(1) - q.tot)
    rv_sat = q_sat / (DT(1) - q.tot)
    S      = q_vap/q_sat - DT(1)

    ag, bg = 5.4 * 1e2, 2.55 * 1e5
    G = DT(1) / (ag + bg / p / rv_sat) / ρ

    av, bv = 1.6, 124.9
    F = av * (ρ/DT(1e3))^DT(0.525)  * rr^DT(0.525) + bv * (ρ/DT(1e3))^DT(0.7296) * rr^DT(0.7296)

    return DT(1) / (DT(1) - q.tot) * S * F * G
end

# example values
q_liq_range  = range(1e-8, stop=5e-3, length=100)
q_ice_range  = range(1e-8, stop=5e-3, length=100)
q_rain_range = range(1e-8, stop=5e-3, length=100)
q_snow_range = range(1e-8, stop=5e-3, length=100)
ρ_air, ρ_air_ground = 1.2, 1.22
q_liq, q_ice, q_tot = 5e-4, 5e-4, 20e-3

plot( q_rain_range * 1e3, [terminal_velocity(param_set, rain_param_set, ρ_air, q_rai) for q_rai in q_rain_range],     linewidth=3, xlabel="q_rain or q_snow [g/kg]", ylabel="terminal velocity [m/s]", label="Rain-CLIMA")
plot!(q_snow_range * 1e3, [terminal_velocity(param_set, snow_param_set, ρ_air, q_sno) for q_sno in q_snow_range],     linewidth=3, label="Snow-CLIMA")
plot!(q_rain_range * 1e3, [terminal_velocity_empirical(q_rai, q_tot, ρ_air, ρ_air_ground) for q_rai in q_rain_range], linewidth=3, label="Rain-Empirical")

T = 273.15
plot( q_liq_range * 1e3, [conv_q_liq_to_q_rai(rain_param_set, q_liq) for q_liq in q_liq_range], linewidth=3, xlabel="q_liq or q_ice [g/kg]", ylabel="autoconversion rate [1/s]", label="Rain")
plot!(q_ice_range * 1e3, [conv_q_ice_to_q_sno(param_set, ice_param_set, PhasePartition(q_tot, 0., q_ice), ρ_air, T-5)  for q_ice in q_ice_range], linewidth=3, label="Snow T=-5C")
plot!(q_ice_range * 1e3, [conv_q_ice_to_q_sno(param_set, ice_param_set, PhasePartition(q_tot, 0., q_ice), ρ_air, T-10) for q_ice in q_ice_range], linewidth=3, label="Snow T=-15C")
plot!(q_ice_range * 1e3, [conv_q_ice_to_q_sno(param_set, ice_param_set, PhasePartition(q_tot, 0., q_ice), ρ_air, T-15) for q_ice in q_ice_range], linewidth=3, label="Snow T=-25C")

plot( q_rain_range * 1e3, [accretion(param_set, liquid_param_set, rain_param_set, q_liq, q_rai, ρ_air) for q_rai in q_rain_range], linewidth=3, xlabel="q_rain or q_snow [g/kg]", ylabel="accretion rate [1/s]", label="Liq+Rain-CLIMA")
plot!(q_rain_range * 1e3, [accretion(param_set, ice_param_set,    rain_param_set, q_ice, q_rai, ρ_air) for q_rai in q_rain_range], linewidth=3, label="Ice+Rain-CLIMA")
plot!(q_snow_range * 1e3, [accretion(param_set, liquid_param_set, snow_param_set, q_liq, q_sno, ρ_air) for q_sno in q_snow_range], linewidth=3, label="Liq+Snow-CLIMA")
plot!(q_snow_range * 1e3, [accretion(param_set, ice_param_set,    snow_param_set, q_ice, q_sno, ρ_air) for q_sno in q_snow_range], linewidth=4, linestyle=:dash, label="Ice+Snow-CLIMA")
plot!(q_rain_range * 1e3, [accretion_empirical(q_rai, q_liq, q_tot) for q_rai in q_rain_range], linewidth=3, label="Liq+Rain-Empirical")

q_ice = 1e-6
plot(q_rain_range * 1e3, [accretion_rain_sink(param_set, ice_param_set, rain_param_set, q_ice, q_rai, ρ_air) for q_rai in q_rain_range], linewidth=3, xlabel="q_rain or q_snow [g/kg]", ylabel="accretion rain sink rate [1/s]", label="q_ice=1e-6")
q_ice = 1e-5
plot!(q_rain_range * 1e3, [accretion_rain_sink(param_set, ice_param_set, rain_param_set, q_ice, q_rai, ρ_air) for q_rai in q_rain_range], linewidth=3, xlabel="q_rain or q_snow [g/kg]", ylabel="accretion rain sink rate [1/s]", label="q_ice=1e-5")
q_ice = 1e-4
plot!(q_rain_range * 1e3, [accretion_rain_sink(param_set, ice_param_set, rain_param_set, q_ice, q_rai, ρ_air) for q_rai in q_rain_range], linewidth=3, xlabel="q_rain or q_snow [g/kg]", ylabel="accretion rain sink rate [1/s]", label="q_ice=1e-4")

q_sno = 1e-6
plot(q_rain_range * 1e3, [accretion_snow_rain(param_set, rain_param_set, snow_param_set, q_rai, q_sno, ρ_air) for q_rai in q_rain_range], linewidth=3, xlabel="q_rain [g/kg]", ylabel="snow-rain accretion rate [1/s] T>0", label="q_snow=1e-6")
q_sno = 1e-5
plot!(q_rain_range * 1e3, [accretion_snow_rain(param_set, rain_param_set, snow_param_set, q_rai, q_sno, ρ_air) for q_rai in q_rain_range], linewidth=3, label="q_snow=1e-5")
q_sno = 1e-4
plot!(q_rain_range * 1e3, [accretion_snow_rain(param_set, rain_param_set, snow_param_set, q_rai, q_sno, ρ_air) for q_rai in q_rain_range], linewidth=3, label="q_snow=1e-4")

q_rai = 1e-6
plot(q_snow_range * 1e3, [accretion_snow_rain(param_set, snow_param_set, rain_param_set, q_sno, q_rai, ρ_air) for q_sno in q_snow_range], linewidth=3, xlabel="q_snow [g/kg]", ylabel="snow-rain accretion rate [1/s] T<0", label="q_rain=1e-6")
q_rai = 1e-5
plot!(q_snow_range * 1e3, [accretion_snow_rain(param_set, snow_param_set, rain_param_set, q_sno, q_rai, ρ_air) for q_sno in q_snow_range], linewidth=3, label="q_rain=1e-5")
q_rai = 1e-4
plot!(q_snow_range * 1e3, [accretion_snow_rain(param_set, snow_param_set, rain_param_set, q_sno, q_rai, ρ_air) for q_sno in q_snow_range], linewidth=3, label="q_rain=1e-4")

# example values
T, p = 273.15 + 15, 90000.
ϵ = 1. / molmass_ratio(param_set)
p_sat = saturation_vapor_pressure(param_set, T, Liquid())
q_sat = ϵ * p_sat / (p + p_sat * (ϵ - 1.))
q_rain_range = range(1e-8, stop=5e-3, length=100)
q_tot = 15e-3
q_vap = 0.15 * q_sat
q_ice = 0.
q_liq = q_tot - q_vap - q_ice
q = PhasePartition(q_tot, q_liq, q_ice)
R = gas_constant_air(param_set, q)
ρ = p / R / T

plot(q_rain_range * 1e3,  [evaporation_sublimation(param_set, rain_param_set, q, q_rai, ρ, T) for q_rai in q_rain_range], xlabel="q_rain [g/kg]", linewidth=3, ylabel="rain evaporation rate [1/s]", label="ClimateMachine")
plot!(q_rain_range * 1e3, [rain_evap_empirical(q_rai, q, T, p, ρ) for q_rai in q_rain_range], linewidth=3, label="empirical")

# example values
T, p = 273.15 - 15, 90000.
ϵ = 1. / molmass_ratio(param_set)
p_sat = saturation_vapor_pressure(param_set, T, Ice())
q_sat = ϵ * p_sat / (p + p_sat * (ϵ - 1.))
q_snow_range = range(1e-8, stop=5e-3, length=100)
q_tot = 15e-3
q_vap = 0.15 * q_sat
q_liq = 0.
q_ice = q_tot - q_vap - q_ice
q = PhasePartition(q_tot, q_liq, q_ice)
R = gas_constant_air(param_set, q)
ρ = p / R / T

plot(q_snow_range * 1e3,  [evaporation_sublimation(param_set, snow_param_set, q, q_sno, ρ, T) for q_sno in q_snow_range], xlabel="q_snow [g/kg]", linewidth=3, ylabel="snow deposition sublimation rate [1/s]", label="T<0")

T, p = 273.15 + 15, 90000.
ϵ = 1. / molmass_ratio(param_set)
p_sat = saturation_vapor_pressure(param_set, T, Ice())
q_sat = ϵ * p_sat / (p + p_sat * (ϵ - 1.))
q_snow_range = range(1e-8, stop=5e-3, length=100)
q_tot = 15e-3
q_vap = 0.15 * q_sat
q_liq = 0.
q_ice = q_tot - q_vap - q_ice
q = PhasePartition(q_tot, q_liq, q_ice)
R = gas_constant_air(param_set, q)
ρ = p / R / T

plot!(q_snow_range * 1e3,  [evaporation_sublimation(param_set, snow_param_set, q, q_sno, ρ, T) for q_sno in q_snow_range], xlabel="q_snow [g/kg]", linewidth=3, ylabel="snow deposition sublimation rate [1/s]", label="T>0")

T=273.15
plot( q_snow_range * 1e3,  [snow_melt(param_set, snow_param_set, q_sno, ρ, T+2) for q_sno in q_snow_range], xlabel="q_snow [g/kg]", linewidth=3, ylabel="snow melt rate [1/s]", label="T=2C")
plot!(q_snow_range * 1e3,  [snow_melt(param_set, snow_param_set, q_sno, ρ, T+4) for q_sno in q_snow_range], xlabel="q_snow [g/kg]", linewidth=3, label="T=4C")
plot!(q_snow_range * 1e3,  [snow_melt(param_set, snow_param_set, q_sno, ρ, T+6) for q_sno in q_snow_range], xlabel="q_snow [g/kg]", linewidth=3, label="T=6C")