Stochastic variability of mass flux in a cloud resolving simulation

(1)

Stochastic variability of mass flux in a cloud

resolving simulation

Jahanshah Davoudi

Thomas Birner, Norm McFarlane and Ted Shepherd

(2)

References

Brenda G. Cohen and George C. Craig, 2004: Quart. J. Roy. Meteor. Soc.

130 , 933-944.

Brenda G. Cohen and George C. Craig, 2006: J. Atmos. Sci.

63 , 1996-2004.

Brenda G. Cohen and George C. Craig, 2006: J. Atmos. Sci.

63 , 2005-2015.

(3)

Convective parameterization

Many clouds and especially the processes within them are

subgrid-scale size

both horizontally and vertically and thus must be parameterized.

This means a mathematical model is constructed that attempts to assess their

(4)

Parameterization Basics

Arakawa

_&

Schubert 1974

Key equilibrium assumption:

(5)

Fluctuations in radiative-convective equilibrium

For convection in equilibrium with a given forcing, the mean mass flux

should be well defined.

At a particular time, this mean value would only be measured in an

infinite

domain

.

For a region of

finite size

:

What is the magnitude and distribution of variability?

(6)

Main assumptions

Assume:

1. Large-scale constraints

- mean mass flux within a region

_h

M

_i

is given in terms

of large scale resolved conditions

2. Scale separation

- environment sufficiently uniform in time and space to

average over a large number of clouds

3. Weak interactions

- clouds feel only mean effects of total cloud field( no

organization)

(7)

Large scale constraint

h

M

_i

is determined by the requirement that the convection balance the large

scale forcing when averaged over a large region.

h

m

_i

is

not

necessarily a function of large scale forcing

Observations suggest that

_h

m

_i

is independent of large scale forcing

Response to the change in forcing is to change the

number

of clouds.

h

m

_i

might be only sensitive to the initial perturbation triggering it and the

dynamical entrainment processes.

(8)

mass flux of individual clouds are statistically un-correlated :

P

_M

(

n

) =

P rob

_{

N

[(0

, M

]) =

n

_}

=

(

λM

)

n

_e

−

λM

n

!

n

= 0

,

1 ,

· · ·

given

λ

= 1

/

(

_h

m

_i

) =

_h

h

_M

N

i

_i

is fixed.

Poisson point process implies:

P

(

m

) =

1 h

m

_i

e

−

m hmi

The total Mass flux for a given

N

Poisson distributed plumes is a

Compound point process

:

M

=

N

X

i

=0

(9)

Predicted distribution

So the Generating function of

M

is calculated exactly:

h

e

tM

_i

=

e

−

Λ

e

Λ

G

(

t

)

(1)

G

(

t

) =

_h

e

tm

_i

Λ =

_h

N

_i

Therefore the probability distribution of the

total mass flux

is exactly given by:

P

(

M

) =

P

(

M

) =

„

h

N

_i

h

m

_i

«

1 /

2 e

−h

N

i

M

−

1 /

2 e

−

M/

h

m

i

I

1

2 „

h

N

_i

h

m

_i

M

«

1 /

2 !

All the moments of

M

are analytically tractable and are functions of

_h

N

_i

and

_h

m

_i

.

h

(

δM

)

2 _i

h

M

_i

2 =

2 h

N

_i

h

M

r

_|

N

_i

=

(

N

+

r

−

1)!

(

N

₋

1)!

h

m

i

r

(10)

Estimates

In a region with area

A

and grid size

∆

x

_≫

L

where

L

the

mean cloud

spacing

is:

L

= (

A/

_h

N

_i

)

1 /

2 = (

_h

m

_i

A/

_h

M

_i

)

1 /

2 Assume latent heat release balance radiative cooling

S

,

rate of Latent heating

≃

Convective mass flux

×

Typical water vapor mass mixing

ratio

q

l

v

q

h

M

_A

i

=

S

Estimate:

S

= 250

W m

−

2 ,

q

= 10

gkg

−

1 and

l

_v

= 2

.

5 _×

10

6 Jkg

−

1 gives

h

M

_i

/A

= 10

−

2 kgs

−

1 m

−

2 h

m

_i

=

wρσ

with

w

_≃

10 ms

−

1 and

σ

_≃

1 km

2 gives

h

m

_{i ≃}

10

7 kgs

−

1 hence

L

_≃

30 km

δM

M

=

√

2 _∆

L

_x

(11)

Simulations with a ’cloud resolving’ model

Resolution:

2km

_×

2km

_×

90 levels

Domain:

96 km

_×

96 km

_×

30 km

Boundary conditions:

doubly periodic, fixed SST of 300 K

Forcing:

An-elastic equations with fully interactive radiation

(12)

(13)

Stochastic variability of up-draught M and N

0 0.002 0.004 0.006 0.008 0.01 0.012 124.5 125 125.5 126 126.5 127 127.5 M(t) (x 10 10 Kg /sec) t(days) z=4989.53 m 0 2 4 6 8 10 12 14 16 124.5 125 125.5 126 126.5 127 127.5 N(t) t(days) z=1071.34 m 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 124.5 125 125.5 126 126.5 127 127.5 mc (t)(x 10 10 Kg /sec) t(days)

(14)

Auto-correlation of up-draught M

-0.2

0

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

60

70 <M(t+

τ

)M(t)>

|

τ

(hours)|

z=2976.26 m

z=6907.73 m

z=9900.00 m

z=12900.00 m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.5

1

1.5

2

2.5

3 <M(t+

τ

)M(t)>

|

τ

(hours)|

z=2976.26 m

z=6907.73 m

z=9900.00 m

z=12900.00 m

(15)

Auto-correlation of up-draught N

-0.2

0

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

60

70 <N(t+

τ

)N(t)>

|

τ

(hours)|

z=2976.26 m

z=6907.73 m

z=9900.00 m

z=12900.00 m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.5

1

1.5

2

2.5

3 <N(t+

τ

)N(t)>

|

τ

(hours)|

z=2976.26 m

z=6907.73 m

z=9900.00 m

z=12900.00 m

(16)

Auto-correlation of up-draught

m

_c

-0.2

0

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

60

70 <m

c

(t+

τ

)m

c

(t)>

|

τ

(hours)|

z=2976.26 m

z=6907.73 m

z=9900.00 m

z=12900.00 m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.2

0.4

0.6

0.8

1 <m

c

(t+

τ

)m

c

(t)>

|

τ

(hours)|

z=2976.26 m

z=6907.73 m

z=9900.00 m

z=12900.00 m

(17)

Comparison of short de-correlation time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.5

1

1.5

2 Auto-correlation

|

τ

(hours)|

w

M

N

-0.2

0

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

60

70 Auto-correlation

|

τ

(hours)|

w

M

N

(18)

Mean characteristics

0

5

10

15

20

25

0

0.001

0.002

0.003

0.004

0.005

0.006 z (Km)

<M> (Kg m

-2

s

-1

)

0

5

10

15

20

25

0

1

2

3

4

5

6 z (Km)

<N>

(19)

Variance scaling

0

2

4

6

8

10

12

14

0

2

4

6

8

10

12 (<M

2

>-<M>

2

)/<M>

2

2/<N>

0

2

4

6

8

10

12

14

0

5

10

15

20

25 <N>

σ

2_N

The scaling of the variance of total mass flux and number of active grids in different heights with the

Craig and Cohen prediction:

h

(

δM

)

2 _i

h

M

_i

2 =

2 h

N

_i

(20)

Skewness scaling

0

2

4

6

8

10

12

0

1

2

3

4

5

6

7

8

9 z (Km)

〈

(M -

〈

M

〉

)

3

〉

⁄

〈

M

〉

3

6 ⁄

〈

N

〉

2

The prediction:

h

(

δM

)

3 _i

h

M

_i

3 =

6 h

N

_i

2

(21)

Altitude variability of total Mass flux PDF

z

= 1517

m

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 20 40 60 80 100 120 140 160 M P(M;z) Measured PDF Free 〈 M 〉 and 〈 N 〉

P(M;z) with measured 〈 M 〉 and 〈 N 〉

z

= 2976

.

26 m

0 0.005 0.01 0.015 0.02 0.025 0 20 40 60 80 100 120 140 M (Kg M−2 s−1) P(M;z) Measured PDF Free 〈 M 〉 and 〈 N 〉

0 2 4 6 8 10 12 14 16 18 20 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 N P(N;z) Measured PDF Fit 〈 N 〉=5.288

Poisson PDF with measured 〈 N 〉=5.28

0 2 4 6 8 10 12 14 16 18 20 0 0.05 0.1 0.15 0.2 0.25 N P(N;z) Measured PDF Fit 〈 N 〉=3.5

(22)

P(M,z) and P(N,z)

z

= 4989

.

5 m

0 0.005 0.01 0.015 0.02 0.025 0.03 0 20 40 60 80 100 120 140 M (Kg M−2 s−1) P(M;z) Measured PDF Free 〈 M 〉 and 〈 N 〉

z

= 6907

.

73 m

0 0.005 0.01 0.015 0.02 0.025 0.03 0 50 100 150 200 250 M (Kg M−2 s−1) P(M;z) Measured PDF Free 〈 M 〉 and 〈 N 〉

0 2 4 6 8 10 12 14 16 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 N P(N;z) Measured PDF Fit 〈 N 〉=2.6

0 5 10 15 20 25 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 N P(N;z) Measured PDF Fit 〈 N 〉=2.45

(23)

P(M,z) and P(N,z)

z

= 8701

.

13 m

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 50 100 150 200 250 M (Kg M−2 s−1) P(M;z) Measured PDF Free 〈 M 〉 and 〈 N 〉

z

= 9900

.

00 m

0 0.005 0.01 0.015 0.02 0.025 0.03 0 50 100 150 200 250 300 350 M (Kg M−2 s−1) P(M;z) Measured PDF Free 〈 M 〉 and 〈 N 〉

0 5 10 15 20 25 30 35 40 0 0.05 0.1 0.15 0.2 0.25 N P(N;z) Measured PDF Fit 〈 N 〉=2.9

0 5 10 15 20 25 30 35 40 45 50 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 N P(N;z) Measured PDF Fit 〈 N 〉=1.7

(24)

P(M,z) and P(N,z)

z

= 11400

.

00 m

0 0.005 0.01 0.015 0.02 0.025 0.03 0 50 100 150 200 250 300 350 400 450 500 M (Kg M−2 s−1) P(M;z) Measured PDF Free 〈 M 〉 and 〈 N 〉

z

= 12900

.

00 m

0 0.005 0.01 0.015 0.02 0.025 0 100 200 300 400 500 600 700 M (Kg M−2 s−1) P(M;z) Measured PDF Free 〈 M 〉 and 〈 N 〉

0 10 20 30 40 50 60 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 N P(N;z) Measured PDF Fit 〈 N 〉=0.95

(25)

Conditional average

h

M

|

N

i

0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5 < M | N>/<m> N/<N> z=1516.91 m <N>*x 0 2 4 6 8 10 12 14 16 18 0 1 2 3 4 5 6 < M | N>/<m> N/<N> z=4989.53 m <N>*x -5 0 5 10 15 20 25 30 35 40 0 2 4 6 8 10 12 < M | N>/<m> N/<N> z=8403.16 m <N>*x -10 0 10 20 30 40 50 60 0 2 4 6 8 10 12 14 16 < M | N>/<m> N/<N> z=11400.00 m <N>*x

h

M

_|

N

_i

=

_h

m

_i

N

(26)

Cross-correlation of M and N

0

2

4

6

8

10

12

14

16

0.7

0.75

0.8

0.85

0.9

0.95

1 z (Km)

<(M-<M>)(N-<N>)>/

σ

_M

σ

_N

The prediction:

h

(

M

_{− h}

M

_i

)(

N

_{− h}

N

_i

)

_i

σ

_M

σ

_N

=

1 √

2

(27)

Low order mixed moment

0

2

4

6

8

10

12

14

16

0

1

2

3

4

5

6

7 z (Km)

<(M-<M>)

2 (N-<N>)>

⁄

σ

_M

2 σ

_N

1 ⁄

<N>

The prediction:

h

(

M

_{− h}

M

_i

)

2 (

N

_{− h}

N

_i

)

_i

σ

2 _M

σ

_N

=

1 √

N

(28)

Fat tails of marginal PDF of active grids

1e-05 1e-04 0.001 0.01 0.1 1 0 5 10 15 20 P(N;z) N z=1516 m z=2641 m z=4129.88 m 1e-05 1e-04 0.001 0.01 0.1 1 0 10 20 30 40 50 60 P(N;z) N z=4129.88 m z=5830 m z=7219 m z=8403 m z=9600 m z=10800 m 1e-05 1e-04 0.001 0.01 0.1 1 0 10 20 30 40 50 60 P(N;z) N z=10800 m z=12000 m z=13200 m z=14400 m z=15600 m

(29)

PDF of dM/dz

1e-06 1e-05 1e-04 0.001 0.01 0.1 1 7e-05 5e-05 3e-05 1e-05 0 -1e-05 -3e-05 P(dM/dz) dM/dz (x107 Kg/ms) [0,1] Km 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1 1e-05 0 -1e-05 P(dM/dz) dM/dz (x107 Kg/ms) [1,4.9] Km 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1 1e-05 0 -1e-05 P(dM/dz) dM/dz (x107 Kg/ms) [4,9,9.9] Km 1e-06 1e-05 1e-04 0.001 0.01 0.1 1 1e-05 0 -1e-05 P(dM/dz) dM/dz (x107 Kg/ms) [9.9,16] Km

(30)

Regime change

2

4

6

8

10

12

0

0.001

0.002

0.003

0.004

0.005

0.006 z (Km)

<m> (Kg m

-2

s

-1

),<M> (Kg m

-2

s

-1

),<N> (x10

3 )

<m>

<M>

<N>

z

_∼

[1

,

6]

Km: The

_h

m

_i

is increasing while

_h

N

_i

is decreasing.

z

_∼

[6

,

11]

Km:The

_h

m

_i

is decreasing while

_h

N

_i

is increasing.

z

_∼

[11

,

15]

Km: The

_h

m

_i

is increasing while

_h

N

_i

is decreasing.

(31)

Cumulative Probability of

q

_l

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 P> (ql ;z) q_l (x10−6) z=1516.91 m z=2641.41 m z=4129.88 m z=5418.46 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 70 80 90 P> (ql ;z) q_l (x10−6) z=5830 m z=7812.87 m z=9000.32 m z=10800 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 70 80 90 100 P> (q l ;z) q_l (x10−6) z=11400.00 m z=12600.00 m z=13800.00 m z=15300.00 m

(32)

Distribution of CAPE and Level of Neutral Buoyancy

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

500 1000

1500

2000

2500

3000

P(CAPE)

CAPE (J Kg

-1

)

0

0.05

0.1

0.15

0.2

0.25

0.3 10500 11000 11500 12000 12500 13000 13500

P(LNB)

LNB (m)

(33)

Size distribution of clouds

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 3 4 5 6 7 8 9 10 P(A) A ( Grid cell) z=2976 m <A>≅ 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 1 2 3 4 5 6 7 8 9 10 P(A) A ( Grid cell) z=8403 m <A>≅ 2 0 0.05 0.1 0.15 0.2 0.25 0.3 1 2 3 4 5 6 7 8 9 10 P(A) A ( Grid cell) z=10500 m <A>≅ 3

(34)

Exponential distribution P(M,z) above LNB

z

= 11400

m

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 50 100 150 200 250 300 350 400 450 500 M (Kg M−2 s−1) P(M;z) Measured PDF Fit 〈 M 〉=0.0017 and 〈 N 〉=1.021 Exponential fit 〈 m 〉=0.0027 Measured 〈 M 〉=0.002 and 〈 N 〉=3.41

z

= 12900

.

00 m

0 0.005 0.01 0.015 0.02 0.025 0 100 200 300 400 500 600 700 M (Kg M−2 s−1) P(M;z) Measured PDF Fit 〈 M 〉=0.00066 and 〈 N 〉=0.5818 Exponential fit 〈 m 〉=0.00152 Measured 〈 M 〉=0.000868 and 〈 N 〉=1.82 0 10 20 30 40 50 60 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 N P(N;z) Measured PDF Fit 〈 N 〉=0.95

Poisson PDF with estimated cluster 〈 N 〉=3.4/3

(35)

Stochastic variability of down-draught M and N

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 124.5 125 125.5 126 126.5 127 127.5 -m d (t) (x 10 10 Kg /sec) t(days) z=4989.53 m 0 2 4 6 8 10 12 124.5 125 125.5 126 126.5 127 127.5 N( d t) t(days) z=4989.53 m 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 124.5 125 125.5 126 126.5 127 127.5 -m d (t) (x 10 10 Kg /sec) t(days)

(36)

Down draft characteristics

0 2 4 6 8 10 12 14 16 18 20 0.0012 0.0008 0.0004 z (Km) -<M_d> ( kg m-2 s-1) 0 5 10 15 20 0 0.5 1 1.5 2 2.5 z (Km) <N> 0 5000 10000 15000 20000 25000 -0.002-0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 z <M> (x 1010 kg/m s)

(37)

Cross-correlation of down-draught

M

_d

and

N

_d

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2

4

6

8

10

12

14

16

18 <M

u

(z) M

d

(z)>

z (Km)

0

0.1

0.2

0.3

0.4

0.5

0.6

2

4

6

8

10

12

14

16

18 <N

u

(z) N

d

(z)>

z (Km)

(38)

P

_d

(

M

;

z

)

and

P

_d

(

N

;

z

)

0 200 400 600 800 1000 1200 1400 1600 1800 0.005 0.004 0.003 0.002 0.001 Pd (M;z) M (x 1010 kg/m s) z=1516.91 m 0 200 400 600 800 1000 1200 0.005 0.003 0.001 Pd (M;z) M (x 1010 kg/m s) z=2976.26 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 12 14 Pd (N;z) N z=1516.91 m 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 5 10 15 20 25 Pd (N;z) N z=2976.26 m

(39)

P

_d

(

M

;

z

)

and

P

_d

(

N

;

z

)

0 100 200 300 400 500 600 700 800 0.005 0.003 0.001 Pd (M;z) M (x 1010 kg/m s) z=4989.53 m 0 100 200 300 400 500 600 700 800 900 1000 0.004 0.003 0.002 0.0005 Pd (M;z) M (x 1010 kg/m s) z=6907.73 m 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 2 4 6 8 10 12 14 16 18 Pd (N;z) N z=4989.53 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 2 4 6 8 10 12 14 16 Pd (N;z) N z=6907.73 m

(40)

P

_d

(

M

;

z

)

and

P

_d

(

N

;

z

)

0 100 200 300 400 500 600 0.004 0.003 0.002 0.0005 Pd (M;z) M (x 1010 kg/m s) z=8403.16 m 0 100 200 300 400 500 600 700 0.004 0.003 0.002 0.0005 Pd (M;z) M (x 1010 kg/m s) z=9900 m 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 1 2 3 4 5 6 7 8 9 10 Pd (N;z) N z=8403 m 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 1 2 3 4 5 6 7 8 9 10 Pd (N;z) N z=9900 m

(41)

P

_d

(

M

;

z

)

and

P

_d

(

N

;

z

)

0 100 200 300 400 500 600 700 800 900 1000 0.004 0.003 0.002 0.0005 Pd (M;z) M (x 1010 kg/m s) z=11400 m 0 50 100 150 200 250 300 350 400 450 0.004 0.003 0.002 0.0005 Pd (M;z) M (x 1010 kg/m s) z=14400 m 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 2 4 6 8 10 12 14 16 18 20 Pd (N;z) N z=12900 m 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 2 4 6 8 10 12 14 16 18 20 Pd (N;z) N z=14400 m

(42)

Tails of

P

_d

(

N

;

z

)

1e-05 1e-04 0.001 0.01 0.1 1 0 5 10 15 20 25 Pd (N;z) N z=876.38 m z=1516.91 m z= 2976.26 m 1e-05 1e-04 0.001 0.01 0.1 1 0 5 10 15 20 25 Pd (N;z) N z=4989.53 m z=6907.73 m z= 8403.16 m 1e-05 1e-04 0.001 0.01 0.1 1 0 5 10 15 20 25 30 35 40 45 Pd (N;z) N z=8403.16 m z=9900 m z= 11400 m 1e-05 1e-04 0.001 0.01 0.1 1 0 5 10 15 20 25 30 35 40 45 Pd (N;z) N z=12900 m z=14400 m z= 15900 m