On models of stochastic condensation in clouds

On models of stochastic condensation in clouds

Gustavo Abade

University of Warsaw · Institute of Geophysics · Faculty of Physics

Atmospheric Physics Seminar March 5, 2021

condensation cloud

condensation level

coagulation Microphysical processes

Cloud condensation nuclei (CCN)

condensation cloud

condensation level

coagulation Microphysical processes

Cloud condensation nuclei (CCN)

Narrow size distribution - stable cloud

r

size distribution

Broad size distribution - unstable cloud

r

size distribution

Narrow size distribution - stable cloud

r

size distribution

Broad size distribution - unstable cloud

r

size distribution

Which is the most likely distribution?

Maximum entropy principle

Liu, Atmos. Res., 12 (1995); Yano, J.-I., JAS, 76 (2019)

Which is the most likely distribution?

Maximum entropy principle

Liu, Atmos. Res., 12 (1995); Yano, J.-I., JAS, 76 (2019)

Which is the most likely distribution?

Maximum entropy principle

Liu, Atmos. Res., 12 (1995); Yano, J.-I., JAS, 76 (2019)

Closed cloud parcel

Given:

I N - number of droplets

I M - total mass of liquid water

Which is the most likely distribution f (r)?

Closed cloud parcel Given:

I N - number of droplets

I M - total mass of liquid water

Which is the most likely distribution f (r)?

Closed cloud parcel Given:

I N - number of droplets

I M - total mass of liquid water

Which is the most likely distribution f (r)?

Closed cloud parcel The most likely f maximizes the spectral entropy:

H =− Z

f (x)[ln f (x)] dx

x - droplet mass

Liu, Atmos. Res., 12 (1995); Yano, J.-I., JAS, 76 (2019)

Closed cloud parcel The most likely f maximizes the spectral entropy:

H =− Z

f (x)[ln f (x)] dx

x - droplet mass

+ constraints

Liu, Atmos. Res., 12 (1995); Yano, J.-I., JAS, 76 (2019)

Closed cloud parcel The most likely f maximizes the spectral entropy:

H =− Z

f (x)[ln f (x)] dx

x - droplet mass

f (x)dx = f (r)dr

Liu, Atmos. Res., 12 (1995); Yano, J.-I., JAS, 76 (2019)

Weibull distribution

Closed cloud parcel f (r)∼ r²exp(−βr³)

r f

Liu, Atmos. Res., 12 (1995)

Which is the least likely distribution?

Maximum energy principle

Which is the least likely distribution?

Maximum energy principle

Closed cloud parcel The least likely f maximizes the populational energy:

E = Elatent+ Esurface+ . . .

I constraints

Liu, Atmos. Res., 12 (1995)

Closed cloud parcel The least likely f maximizes the populational energy:

E = Elatent+ Esurface+ . . .

I constraints

Liu, Atmos. Res., 12 (1995)

Monodisperse cloud

Closed cloud parcel f (r)∼ δ(r − hri)

f

hri r

Liu, Atmos. Res., 12 (1995)

Most likely

r f

Least likely

f

hri r

Many issues: - “no dynamics”

- equilibrium×non-equilibrium - closed system× open system

Most likely

r f

Least likely

f

hri r

Many issues:

- “no dynamics”

- equilibrium×non-equilibrium - closed system× open system

Most likely

r f

Least likely

f

hri r

Many issues:

- “no dynamics”

- equilibrium×non-equilibrium - closed system× open system

Most likely

r f

Least likely

f

hri r

Many issues:

- “no dynamics”

- equilibrium×non-equilibrium

- closed system× open system

Most likely

r f

Least likely

f

hri r

Many issues:

- “no dynamics”

- equilibrium×non-equilibrium - closed system× open system

Let us describe the condensation process

Diffusional growth

Droplet growth dr dt = 1

r DhSi

hSi - mean-field supersaturation

LES grid box

r

water droplet moist air

∆∈ inertial range

Equation for f (r, t)

∂f

∂t =− ∂

∂r(˙rf ) +· · ·

Advection in radius space with velocity

˙r = DhSi r

Narrow size distribution!

t = 0

t = 10³ s

f

r

Equation for f (r, t)

∂f

∂t =− ∂

∂r(˙rf ) +· · ·

Advection in radius space with velocity

˙r = DhSi r

Narrow size distribution!

t = 0

t = 10³ s

f

r

Equation for f (r, t)

∂f

∂t =− ∂

∂r(˙rf ) +· · ·

Advection in radius space with velocity

˙r = DhSi r

Narrow size distribution!

t = 0

t = 10³ s

f

r

Eulerian stochastic model

Numerical simulation

Lasher-Trapp et al., QJRMS, 131 (2005)

Kinetic equation for f (r; x, t)

∂f

∂t +∇ · (uf)=−∂

∂r( ˙rf ) +· · · ˙r = DS r

x ∆

Kinetic equation for f (r; x, t)

∂f

∂t +∇ · (uf)=−∂

∂r( ˙rf ) +· · · ˙r = DS r

x ∆

Kinetic equation for f (r; x, t)

∂f

∂t +∇ · (uf)=−∂

∂r( ˙rf ) +· · · ˙r = DS r

x ∆

dependent

variable = mean + fluctuation

Kinetic equation for f (r; x, t)

∂f

∂t +∇ · (uf)=−∂

∂r( ˙rf ) +· · · ˙r = DS r

x ∆

f =hfi + f⁰

u =hui + u⁰

˙r =h ˙ri + ˙r⁰

S =hSi + S⁰

∂hfi

∂t +∇ · [huihfi] = − ∂

∂r[h ˙rihfi]

+ ∇ ·hu⁰f⁰i+ ∂

∂rh ˙r⁰f⁰i+· · ·

Mean growth rate (narrowing):

h ˙ri = DhSi r

Turbulent effect (broadening):

hu⁰f⁰i = ? h ˙r⁰f⁰i = ?

∂hfi

∂t +∇ · [huihfi] = − ∂

∂r[h ˙rihfi]

+ ∇ ·hu⁰f⁰i+ ∂

∂rh ˙r⁰f⁰i+· · · Mean growth rate (narrowing):

h ˙ri = DhSi r

Turbulent effect (broadening):

hu⁰f⁰i = ? h ˙r⁰f⁰i = ?

∂hfi

∂t +∇ · [huihfi] = − ∂

∂r[h ˙rihfi]

+ ∇ ·hu⁰f⁰i+ ∂

∂rh ˙r⁰f⁰i+· · · Mean growth rate (narrowing):

h ˙ri = DhSi r

Turbulent effect (broadening):

hu⁰f⁰i = ? h ˙r⁰f⁰i = ?

Slow microphysics hu⁰f⁰i = −K∇hfi

K− turbulent diffusivity

Cannot explain the observed spectrum broadening

Buikov, M. V. (1960’s)

Slow microphysics hu⁰f⁰i = −K∇hfi

K− turbulent diffusivity

Cannot explain the observed spectrum broadening

Buikov, M. V. (1960’s)

Slow microphysics hu⁰f⁰i = −K∇hfi

K− turbulent diffusivity

Cannot explain the observed spectrum broadening

Buikov, M. V. (1960’s)

Fast microphysics

hu⁰f⁰i = −K ∇hfi + ?

h ˙r⁰f⁰i = ? + ?

supersaturation

fluctuations ↔ vertical velocity fluctuations

Fast microphysics

hu⁰f⁰i = −K ∇hfi + ?

h ˙r⁰f⁰i = ? + ?

supersaturation

fluctuations ↔ vertical velocity fluctuations

S(t) w(t)

τ∼ 1/nhri

dS dt =−1

τ [S− S^eq(t)] Seq(t) = aw(t)τ

w(t) S(t)

Slow microphysics (τ big) hS⁰w⁰i = 0

Fast microphysics (τ small) S(t)∼ w(t)

S(t) w(t)

τ∼ 1/nhri

dS dt =−S

τ + aw(t)

τ − phase relaxation time

dS dt =−1

τ [S− S^eq(t)] Seq(t) = aw(t)τ

w(t) S(t)

Slow microphysics (τ big) hS⁰w⁰i = 0

Fast microphysics (τ small) S(t)∼ w(t)

S(t) w(t)

τ∼ 1/nhri

dS dt =−1

τ [S− S^eq(t)]

Seq(t) = aw(t)τ

w(t) S(t)

Slow microphysics (τ big) hS⁰w⁰i = 0

Fast microphysics (τ small) S(t)∼ w(t)

S(t) w(t)

τ∼ 1/nhri

dS dt =−1

τ [S− S^eq(t)]

Seq(t) = aw(t)τ

w(t) S(t)

Slow microphysics (τ big) hS⁰w⁰i = 0

Fast microphysics (τ small) S(t)∼ w(t)

S(t) w(t)

τ∼ 1/nhri

dS dt =−1

τ [S− S^eq(t)]

Seq(t) = aw(t)τ

w(t) S(t)

Slow microphysics (τ big) hS⁰w⁰i = 0

Fast microphysics (τ small) S(t)∼ w(t)

S(t) w(t)

τ∼ 1/nhri

dS dt =−1

τ [S− S^eq(t)]

Seq(t) = aw(t)τ

w(t) S(t)

Slow microphysics (τ big) hS⁰w⁰i = 0

Fast microphysics (τ small) S(t)∼ w(t)

Condensation reversibility

S(t)∼ w(t)

B A

h

˙ m∼dr³

dt = C1w(t) m(z = h)− m(z = 0) = C²h

A. Khain et al., Atmos. Res., 55 (2000)

Condensation reversibility

S(t)∼ w(t)

B A

h

˙ m∼dr³

dt = C1w(t) m(z = h)− m(z = 0) = C²h

A. Khain et al., Atmos. Res., 55 (2000)

r

Growth rate

˙r = DhSi r ∼ 1

r

Mass rate

˙ m∼dr³

dt ∼ r² ˙r∼ r

Supersaturation absorption is faster for larger droplets

r

Effective microscale supersaturation

S_eff≈hSi hri⁻¹r

˙reff= DSeff

r ≈DhSi hri

Fluctuation in growth rate

˙r⁰ ≈DS⁰ hri

Suppress the growth of smaller droplets

r

Effective microscale supersaturation

S_eff≈hSi hri⁻¹r

˙reff= DSeff

r ≈DhSi hri Fluctuation in growth rate

˙r⁰ ≈DS⁰ hri

Suppress the growth of smaller droplets

Direct Numerical Simulations

S⁰= S− hSi

r⁰ = r− hri

Paoli and Sharif, JAS, 66 (2009)

hu⁰f⁰i = − K∇hfi− K¹∂rhfi

h ˙r⁰f⁰i = − K2∇hfi − K3∂_rhfi

Ki− effective diffusion coefficients

hu⁰f⁰i = − K∇hfi− K¹∂rhfi

h ˙r⁰f⁰i = − K2∇hfi − K3∂rhfi

Ki− effective diffusion coefficients

hu⁰f⁰i = − K∇hfi− K¹∂rhfi

h ˙r⁰f⁰i = − K2∇hfi − K3∂rhfi

Ki− effective diffusion coefficients

Gamma distribution

f (r)∼ r^pexp(−βr)

I power law p∼ 5 − 10 I exponential tail I only one mode!

Khvorostyanov and Curry, JAS, 66 (1999)

Gamma distribution

f (r)∼ r^pexp(−βr)

I power law p∼ 5 − 10

I exponential tail I only one mode!

Khvorostyanov and Curry, JAS, 66 (1999)

Gamma distribution

f (r)∼ r^pexp(−βr)

I power law p∼ 5 − 10 I exponential tail

I only one mode!

Khvorostyanov and Curry, JAS, 66 (1999)

Gamma distribution

f (r)∼ r^pexp(−βr)

I power law p∼ 5 − 10 I exponential tail I only one mode!

Khvorostyanov and Curry, JAS, 66 (1999)

Lagrangian stochastic model

Lagrangian model

hW i W_i^′

S_i

T and pdecreasing

dS_i⁰ dt =−S_i⁰

τ_c − S_i⁰

τ_m+ aW_i⁰(t)

τ_c∼ 1

Nhri (condensation)

τm∼ eddy turnover time (mixing)

I W⁰(t): prescribed stochastic process.

Celani et al., EPL, 70 (2005); Grabowski and Abade, JAS, 74 (2017)

Lagrangian model

hW i W_i^′

S_i

T and pdecreasing

dS_i⁰ dt =−S_i⁰

τ_c − S_i⁰

τ_m+ aW_i⁰(t)

τ_c∼ 1

Nhri (condensation)

τm∼ eddy turnover time (mixing)

I W⁰(t): prescribed stochastic process.

Celani et al., EPL, 70 (2005); Grabowski and Abade, JAS, 74 (2017)

Kinematic framework Synthetic turbulent-like flow

Turbulent-like flow

u = (u, w) = ∂ψ

∂z,−∂ψ

∂x



ψ(r, t) =X

random harmonics

0 200 400 600 800 1000 1200

0 500 1000 1500 2000 2500

z [m]

x [m]

hw²i =σ_w²(z) hw(x⁰, z)w(x⁰+ x, z)i =Cˆw(x)σ²_w(z)

Pinsky et al., JAS, 65 (2008), ..., Magaritz-Ronen et al., ACP, 16 (2016)

Vertical profiles

Size distribution at different heights

I broader distribution in the presence of turbulence

I bimodal distributions

—no turbulence —turbulence (model 1) — turbulence (model 2)

Size distribution at different heights

I broader distribution in the presence of turbulence

I bimodal distributions

—no turbulence —turbulence (model 1) — turbulence (model 2)

Size distribution at different heights

I broader distribution in the presence of turbulence

I bimodal distributions

—no turbulence —turbulence (model 1) — turbulence (model 2)

Size distribution at different heights

—no turbulence —turbulence (model 1) — turbulence (model 2)

Summary

I Problem of narrow × broadsize distributions in clouds

I Subgrid fluctuations in Eulerianmodels are tricky

I Lagrangian approach is more natural

THANK YOU!

Summary

I Problem of narrow × broadsize distributions in clouds

I Subgrid fluctuations in Eulerianmodels are tricky

I Lagrangian approach is more natural

THANK YOU!

Summary

I Problem of narrow × broadsize distributions in clouds

I Subgrid fluctuations in Eulerianmodels are tricky

I Lagrangian approach is more natural

THANK YOU!

Summary

I Problem of narrow × broadsize distributions in clouds

I Subgrid fluctuations in Eulerianmodels are tricky

I Lagrangian approach is more natural

THANK YOU!