stochastic versus uncertainty modelingdswc09/contributions/friederichs... · 2009-09-10 ·...

IntroductionAnalytical Solutions

The Cloud ModelConclusions

Stochastic versus Uncertainty Modeling

Petra Friederichs, Michael Weniger, Sabrina Bentzien,

Andreas Hense

Meteorological Institute, University of Bonn

Dynamics and Statistic in Weather and Climate

Dresden, July 29-31 2009

P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 1 / 21



MotivationOutline

Uncertainty Modeling

I Initial conditions – aleatoric uncertainty

Ensemble with perturbed initial conditions

I Model error – epistemic uncertainty

Perturbed physic and/or multi model ensembles

Simulations solving deterministic model equations!




MotivationOutline

Stochastic Modeling

I Stochastic parameterization – aleatoric uncertainty

Stochastic model ensemble

I Initial conditions – aleatoric uncertainty

Ensemble with perturbed initial conditions

I Model error – epistemic uncertainty

Perturbed physic and/or multi model ensembles

Simulations solving stochastic model equations!




MotivationOutline

Outline

Contrast both concepts

I Analytical solutions of simple damping equation

dv(t) = −µv(t)dt

I Simplified, 1-dimensional, time-dependent cloud model

I Problems




Perturbed PhysicsStochastic ModelConclusions

Equation of Motion

Small particle of mass m with velocity v(t)

Subject to damping force F (t) = −µm · v(t)

dv(t) = −µv(t)dt

Solution of deterministic problem

v(t) = v0 exp(−µt)





’Perturbed Physics’

γ is random variable

dv(t) = −γv(t)dt.

Solutions for different γ0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

t

v

deterministicnormaluniformgamma

γN ∼ N (µ, σ2) E [vN (t)] = v0 exp(− µt +

σ2

2t2)

γU ∼ U(µ±√

3σ)

E [vU (t)] = v0 exp(−µt)sinh(

√3σt)√

3σt

γΓ ∼ Γ(µ2

σ2,σ2

µ

)E [vΓ(t)] = v0

(1− −µt

µ2/σ2

)−µ2/σ2





Stochastic Model

Damping from small-scale stochastic process

dv(t) = −µv(t)dt + v(t)σξdt,

ξt : Langevin force with

E [ξt ] = 0 and Cov[ξs , ξt ] = δ(t − s)

Stratonovich calculus

dvt = −µvtdt + σvt ◦ dWt

Solution

vt = v0 exp(− µt + σWt

)(exponential Brownian motion)





Exponential Brownian motion

E[vt ] = v0 exp(−(µ− σ2

2

)t)

Var(vt

)= v2

0

[exp

(− 2(µ− σ2

)t)− exp

(− 2(µ− σ2

2

)t)]

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Three realisation of an exponential Brownian motion with µ=2, σ2=1

Time t

v t

det. solutionmean value

0 1 2 3 4 5 60

1

2

3

4

5

6Three realisation of an exponential Brownian motion with µ=2, σ2=3

Time t

v t

det. solutionmean value





Non-delta Correlated Gaussian Noise

dv(t) = −µdt + v(t)εtv(t)dt

Solutionvt = v0 exp

(− µt +

∫ t

0εsds

)Noise εt with

E[εt ] = 0, Cov[εs , εt ] =D

2kexp(−k|t − s|).

is an Ornstein-Uhlenbeck process solving

dεt = −kεtdt +√

DdWt





Non-delta Correlated Gaussian Noise

E[vt ] = v0 exp(− µt +

D

2k2

(t − 1− e−kt

k

))

D = 2k D = k2

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

t

v

deterministicdet. normalExpon. BrownianOU(k=0.01)OU(k=1)OU(k=100)

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

t

v

deterministicdet. normalExpon. BrownianOU(k=0.01)OU(k=1)OU(k=100)





D = 2k D = k2





’Perturbed physics’

E [v(t)]→∞ for t →∞ for all parameter distributions with

non-positive support

Stochastic model

Exponential Brownian motion:

I Non-stable solutions (σ2 > 2µ), increasing variance (σ2 > µ)

OU noise:

I Noise induced drift increases with 1/k (D = 2k)

I ’Perturbed physics’ solution for k → 0 (D = 2k)

I Exponential Brownian motion for k →∞ (D = k2)




Model DescriptionPerturbed PhysicsStochastic Parameterization

1-dimensional Cloud Model

I Cylinder - constant radius

I Horizontally homogen in

constant environment

I Prognostic equations for

w(z), T (z), qv (z), qc(z),

qr (z)Fig: Dynamics of Multiscale Earth Systems,

C.Simmer(Eds.)





Koeln 04.07.1994 RR=11,723mm

Zeit in min

Hoe

he in

km

0.2

0.2

0.4

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2

0 10 20 30 40 50 60 70 80 90

0

1

2

3

4

5

6

7

8

9

10

0

10

20

30

40mm/h

g/kg

0

1

2

3

4





0 = −1

ρ

∂(ρw)

∂z−

2

Ru

∂w

∂t=−w

∂w

∂z+

2

Rα2|we − w |(we − w) +

2

R(w − w)u +

„Tv − Tv,e

Tv,e

«g − (qc + qr )g

∂T

∂t=−w

∂T

∂z+

2

Rα2|we − w |(Te − T ) +

2

R(T − T )u − Γdw + SST

∂qv

∂t=−w

∂qv

∂z+

2

Rα2|we − w |(qv,e − qv ) +

2

R(qv − qv )u−

dqv

dt

∂qc

∂t=−w

∂qc

∂z+

2

Rα2|we − w |(qc,e − qc ) +

2

R(qc − qc )u −

dqc

dt

∂qr

∂t=−w

∂qr

∂z| {z }advection

+2

Rα2|we − w |(qr,eqr )| {z }turbulent entrainment

+2

R(qr − qr )u| {z }

dynamical entr.

−dqr

dt+ VR

∂r

∂z+

qr

ρ

∂(ρVR)

∂z| {z }sources





Micro-physical

processes after

Kessler (1969)...

Fig: von der Emde, Kahlig (1989)





Condensation:

qv → qc

− dqv

dt

VC

=qv − qvws

1 + εL2cqvws

cpRaT 2

· 1

∆tfor qv > qvws

Conversion:

qc → qr

− dqc

dt

CRAU

= kc(qc − qc,0) for qc > qc,0

qc,0 = 1ρ , kc = 10−3

Accretion:

qc → qr

− dqc

dt

CRAC

=π

4Eαrn0,rqc · λ−(3+βr ) · Γ(3 + βr )

E = 1 collection efficiency, αr , βr ;n0,r : Marshall-Palmer

λ−(3+βr ) =(

ρqr

πρpsn0,r

) (3+βr )4





Perturbed Physics

kc → kc (1 + η) E → E (1 + η)ra

in in

mm

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

−1 0 1 2

Conversion

rain

in m

m

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

−1 0 1 2

Accretion





Perturbed Physics

rain in mm

dens

ity

0.00

0.01

0.02

0.03

0.04

0.05

11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 13.2 13.4

Conversion

rain in mm

dens

ity0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Accretion

=







Stochastic Parameterization

(solved with Milstein Scheme)

kc → kc (1 + ξt) E → E (1 + ξt)

rain in mm

dens

ity

0.00

0.02

0.04

0.06

0.08

11.64 11.68 11.72 11.76 11.8

Conversion

rain in mm

dens

ity

0.00

0.01

0.02

0.03

0.04

0.05

11.2 11.4 11.6 11.8 12 12.2 12.4

Accretion




Conclusions and Outlook

I ’Perturbed physics’ ensembles do not represent realizations

from real process

I Stochastic parameterization may provide realistic distributions

I Solutions strongly depend on covariance function of noise (in

time and in space)

I Stochastic parameterizations should be derived from

microphysical processes


stochastic versus uncertainty modelingdswc09/contributions/friederichs... · 2009-09-10 ·...

Documents