Stochastic variability of mass flux in a cloud resolving simulation

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(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

2N

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 〉

P(M;z) with measured 〈 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 〉

P(M;z) with measured 〈 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 〉

P(M;z) with measured 〈 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

Poisson PDF with measured 〈 N 〉=2.7

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 〉

P(M;z) with measured 〈 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 〉

P(M;z) with measured 〈 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

Poisson PDF with measured 〈 N 〉=3.6

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 〉

P(M;z) with measured 〈 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 〉

P(M;z) with measured 〈 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

Poisson PDF with measured 〈 N 〉=3.4

0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 N P(N;z) Measured PDF Fit 〈 N 〉=0.5

(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) ql (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) ql (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) ql (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

0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 N P(N;z) Measured PDF Fit 〈 N 〉=0.5

(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) -<Md> ( 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

(43)

Summary

The statistics of the mass flux for the range of heights from cloud base up to

5

Km

matches with the results of the Cohen and Craig (2006).

At very high altitudes above

10

Km

the statistics of the mass flux is mainly

controlled by intermittently penetrating individual plumes which are

exponentially distributed.

Due to the increasing size of a typical cloud the statistics of the number of

active grids deviates from the prediction of Poisson theory for the higher

altitudes from

6

Km

up to

12

Km

.

Accounting for the typical size of the plumes above

10

Km

shows the

consistency of the Poisson statistics with the data.

Low order conditional and joint statistics of

M

and

N

are less compatible

with the predictions of the Cohen and Craig (2006).

Figure

Updating...

References

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