Stochastic variability of mass flux in a cloud resolving simulation

Transcription

1 Stochastic variability of mass flux in a cloud resolving simulation Jahanshah Davoudi Thomas Birner, orm McFarlane and Ted Shepherd Physics department, University of Toronto Stochastic variability of mass flux in a cloud resolving simulation p./43

2 References Brenda G. Cohen and George C. Craig, 24: Quart. J. Roy. Meteor. Soc. 3, Brenda G. Cohen and George C. Craig, 26: J. Atmos. Sci. 63, Brenda G. Cohen and George C. Craig, 26: J. Atmos. Sci. 63, Plant R.S. and George C. Craig, 28: J. Amtos. Sci. 65, Stochastic variability of mass flux in a cloud resolving simulation p.2/43

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 effects in terms of large scale model resolved quantities. Stochastic variability of mass flux in a cloud resolving simulation p.3/43

4 Parameterization Basics Arakawa & Schubert 974 Key equilibrium assumption: τ adj τ ls Stochastic variability of mass flux in a cloud resolving simulation p.4/43

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? What scale must one average over to reduce it to a desired level? Stochastic variability of mass flux in a cloud resolving simulation p.5/43

6 Main assumptions Assume:.Large-scale constraints- mean mass flux within a region M 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) Find the distribution function subject to these constraints Stochastic variability of mass flux in a cloud resolving simulation p.6/43

7 Large scale constraint M is determined by the requirement that the convection balance the large scale forcing when averaged over a large region. m is not necessarily a function of large scale forcing Observations suggest that m is independent of large scale forcing Response to the change in forcing is to change the number of clouds. m might be only sensitive to the initial perturbation triggering it and the dynamical entrainment processes. Stochastic variability of mass flux in a cloud resolving simulation p.7/43

8 mass flux of individual clouds are statistically un-correlated : given λ = /( m ) = M P M (n) = Prob{ [(, M]) = n} = (λm)n e λm Poisson point process implies: is fixed. P(m) = m e m m n! n =,, The total Mass flux for a given Poisson distributed plumes is a Compound point process: M = X i= m i Stochastic variability of mass flux in a cloud resolving simulation p.8/43

9 Predicted distribution So the Generating function of M is calculated exactly: e tm = e Λ e ΛG(t) () G(t) = e tm Λ = Therefore the probability distribution of the total mass flux is exactly given by: P(M) = P(M) = «/2 e M /2 e M/ m I 2 m «! /2 m M All the moments of M are analytically tractable and are functions of and m. (δm) 2 M 2 = 2 M r = ( + r )! m r ( )! Stochastic variability of mass flux in a cloud resolving simulation p.9/43

10 Estimates In a region with area A and grid size x L where L the mean cloud spacing is: L = (A/ ) /2 = ( m A/ M ) /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 M A = S Estimate: S = 25Wm 2, q = gkg and l v = Jkg gives M /A = 2 kgs m 2 m = wρσ with w ms and σ km 2 gives m 7 kgs hence L 3km δm M = 2 L x Stochastic variability of mass flux in a cloud resolving simulation p./43

11 Simulations with a cloud resolving model Resolution: 2km 2km 9 levels Domain: 96 km 96 km 3 km Boundary conditions: doubly periodic, fixed SST of 3 K Forcing: An-elastic equations with fully interactive radiation scheme Stochastic variability of mass flux in a cloud resolving simulation p./43

12 A 2D cut through the convective field Stochastic variability of mass flux in a cloud resolving simulation p.2/43

13 Stochastic variability of up-draught M and M(t) (x Kg /sec).2 z= m (t) 6 z=7.34 m t(days) t(days).4.2 m c (t)(x Kg /sec) t(days) Stochastic variability of mass flux in a cloud resolving simulation p.3/43

14 Auto-correlation of up-draught M <M(t+τ)M(t)> τ (hours) z= m z= m z=99. m z=29. m <M(t+τ)M(t)> τ (hours) z= m z= m z=99. m z=29. m Stochastic variability of mass flux in a cloud resolving simulation p.4/43

15 Auto-correlation of up-draught <(t+τ)(t)> τ (hours) z= m z= m z=99. m z=29. m <(t+τ)(t)> τ (hours) z= m z= m z=99. m z=29. m Stochastic variability of mass flux in a cloud resolving simulation p.5/43

16 Auto-correlation of up-draught m c <m c (t+τ)m c (t)> z= m z= m z=99. m z=29. m <m c (t+τ)m c (t)> z= m z= m z=99. m z=29. m τ (hours) τ (hours) Stochastic variability of mass flux in a cloud resolving simulation p.6/43

17 Comparison of short de-correlation time Auto-correlation τ (hours) w M Auto-correlation τ (hours) w M Stochastic variability of mass flux in a cloud resolving simulation p.7/43

18 Mean characteristics z (Km) 5 z (Km) <M> (Kg m -2 s - ) <> Stochastic variability of mass flux in a cloud resolving simulation p.8/43

19 Variance scaling (<M 2 >-<M> 2 )/<M> 2 2/<> <> σ The scaling of the variance of total mass flux and number of active grids in different heights with the Craig and Cohen prediction: (δm) 2 M 2 = 2 Stochastic variability of mass flux in a cloud resolving simulation p.9/43

20 Skewness scaling 2 8 z (Km) (M - M ) 3 M The prediction: (δm) 3 M 3 = 6 2 Stochastic variability of mass flux in a cloud resolving simulation p.2/43

21 Altitude variability of total Mass flux PDF 6 4 z = 57 m Measured PDF Free M and P(M;z) with measured M and 4 2 z = m Measured PDF Free M and P(M;z) with measured M and 2 8 P(M;z) 8 P(M;z) M M (Kg M 2 s ).8.6 Measured PDF Fit =5.288 Poisson PDF with measured = Measured PDF Fit =3.5 Poisson PDF with measured = P(;z).8 P(;z) Stochastic variability of mass flux in a cloud resolving simulation p.2/43

22 P(M,z) and P(,z) 4 2 z = m Measured PDF Free M and P(M;z) with measured M and 25 2 z = m Measured PDF Free M and P(M;z) with measured M and 8 5 P(M;z) P(M;z) M (Kg M 2 s ) M (Kg M 2 s ).35.3 Measured PDF Fit =2.6 Poisson PDF with measured = Measured PDF Fit =2.45 Poisson PDF with measured = P(;z) P(;z) Stochastic variability of mass flux in a cloud resolving simulation p.22/43

23 P(M,z) and P(,z) 25 2 z = 87.3 m Measured PDF Free M and P(M;z) with measured M and 35 3 z = 99. m Measured PDF Free M and P(M;z) with measured M and P(M;z) P(M;z) M (Kg M 2 s ) M (Kg M 2 s ).25 Measured PDF Fit =2.9 Poisson PDF with measured = Measured PDF Fit =.7 Poisson PDF with measured = P(;z) P(;z) Stochastic variability of mass flux in a cloud resolving simulation p.23/43

24 P(M,z) and P(,z) z = 4. m Measured PDF Free M and P(M;z) with measured M and 7 6 z = 29. m Measured PDF Free M and P(M;z) with measured M and P(M;z) 25 P(M;z) M (Kg M 2 s ) M (Kg M 2 s ).45.4 Measured PDF Fit =.95 Poisson PDF with measured = Measured PDF Fit =.5 Poisson PDF with measured = P(;z).2 P(;z) Stochastic variability of mass flux in a cloud resolving simulation p.24/43

25 Conditional average M < M >/<m> z=56.9 m <>*x < M >/<m> z= m <>*x /<> /<> < M >/<m> z=843.6 m <>*x < M >/<m> z=4. m <>*x /<> /<> M = m Stochastic variability of mass flux in a cloud resolving simulation p.25/43

26 Cross-correlation of M and z (Km) The prediction: <(M-<M>)(-<>)>/σ M σ (M M )( ) σ M σ = 2 Stochastic variability of mass flux in a cloud resolving simulation p.26/43

27 Low order mixed moment z (Km) <(M-<M>) 2 (-<>)> σ M 2 σ <> The prediction: (M M ) 2 ( ) σ 2 M σ = Stochastic variability of mass flux in a cloud resolving simulation p.27/43

28 Fat tails of marginal PDF of active grids P(;z)... z=56 m z=264 m z= m P(;z)... z= m z=583 m z=729 m z=843 m z=96 m z=8 m e-4 e-4 e e z=8 m z=2 m z=32 m z=44 m z=56 m P(;z).. e-4 e Stochastic variability of mass flux in a cloud resolving simulation p.28/43

29 PDF of dm/dz [,] Km [,4.9] Km.. P(dM/dz).. e-4 P(dM/dz).. e-4 e-5 e-5 e-6 e-6-3e-5 -e-5 e-5 3e-5 5e-5 dm/dz (x 7 Kg/ms) 7e-5 e-7 -e-5 dm/dz (x 7 Kg/ms) e-5 [4,9,9.9] Km [9.9,6] Km.. P(dM/dz).. e-4 e-5 P(dM/dz).. e-4 e-6 e-5 e-7 -e-5 dm/dz (x 7 Kg/ms) e-5 e-6 -e-5 dm/dz (x 7 Kg/ms) e-5 Stochastic variability of mass flux in a cloud resolving simulation p.29/43

30 Regime change 2 <m> <M> <> z (Km) <m> (Kg m -2 s - ),<M> (Kg m -2 s - ),<> (x 3 ) z [, 6] Km: The m is increasing while is decreasing. z [6, ] Km:The m is decreasing while is increasing. z [, 5] Km: The m is increasing while is decreasing. Stochastic variability of mass flux in a cloud resolving simulation p.3/43

31 Cumulative Probability of q l z=56.9 m z=264.4 m z= m z= m z=583 m z= m z=9.32 m z=8 m P > (q l ;z) P > (q l ;z) q l (x 6 ) q l (x 6 ) P > (q l ;z) q l (x 6 ) z=4. m z=26. m z=38. m z=53. m Stochastic variability of mass flux in a cloud resolving simulation p.3/43

32 Distribution of CAPE and Level of eutral Buoyancy P(CAPE) P(LB) CAPE (J Kg - ) LB (m) Stochastic variability of mass flux in a cloud resolving simulation p.32/43

33 Size distribution of clouds z=2976 m <A> z=843 m <A> 2 P(A).4.3 P(A) A ( Grid cell) A ( Grid cell).3.25 z=5 m <A> 3.2 P(A) A ( Grid cell) Stochastic variability of mass flux in a cloud resolving simulation p.33/43

34 Exponential distribution P(M,z) above LB z = 4 m Measured PDF Fit M =.7 and =.2 Exponential fit m =.27 Measured M =.2 and = z = 29. m Measured PDF Fit M =.66 and =.588 Exponential fit m =.52 Measured M =.868 and = P(M;z) 25 P(M;z) M (Kg M 2 s ) M (Kg M 2 s ).45.4 Measured PDF Fit =.95 Poisson PDF with estimated cluster =3.4/3.7.6 Measured PDF Fit =.5 Poisson PDF with estimated cluster =.8/ P(;z).2 P(;z) Stochastic variability of mass flux in a cloud resolving simulation p.34/43

35 Stochastic variability of down-draught M and -m d (t) (x Kg /sec).45 z= m t(days) ( d t) 2 z= m t(days) -m d (t) (x Kg /sec) t(days) Stochastic variability of mass flux in a cloud resolving simulation p.35/43

36 Down draft characteristics z (Km) <M d > ( kg m -2 s - ) z (Km) <> z <M> (x kg/m s) Stochastic variability of mass flux in a cloud resolving simulation p.36/43

37 Cross-correlation of down-draught M d and d.8.6 <M u (z) M d (z)> < u (z) d (z)> z (Km) z (Km) Stochastic variability of mass flux in a cloud resolving simulation p.37/43

38 P d (M; z) and P d (; z) z=56.9 m 2 z= m 2 8 P d (M;z) 8 P d (M;z) M (x kg/m s) M (x kg/m s).6 z=56.9 m.35 z= m P d (;z).3 P d (;z) Stochastic variability of mass flux in a cloud resolving simulation p.38/43

39 P d (M; z) and P d (; z) P d (M;z) z= m P d (M;z) z= m M (x kg/m s) M (x kg/m s) z= m z= m P d (;z).25.2 P d (;z) Stochastic variability of mass flux in a cloud resolving simulation p.39/43

40 P d (M; z) and P d (; z) 6 z=843.6 m 7 z=99 m P d (M;z) 3 P d (M;z) M (x kg/m s) M (x kg/m s).8.7 z=843 m.8.7 z=99 m P d (;z).4.3 P d (;z) Stochastic variability of mass flux in a cloud resolving simulation p.4/43

41 P d (M; z) and P d (; z) 9 z=4 m 45 4 z=44 m P d (M;z) P d (M;z) M (x kg/m s) M (x kg/m s)..9 z=29 m.45.4 z=44 m P d (;z) P d (;z) Stochastic variability of mass flux in a cloud resolving simulation p.4/43

42 Tails of P d (; z). z= m z=56.9 m z= m. z= m z= m z= m P d (;z).. P d (;z).. e-4 e-4 e e z=843.6 m z=99 m z= 4 m. z=29 m z=44 m z= 59 m P d (;z).. P d (;z).. e-4 e-4 e e Stochastic variability of mass flux in a cloud resolving simulation p.42/43

43 Summary The statistics of the mass flux for the range of heights from cloud base up to 5Km matches with the results of the Cohen and Craig (26). At very high altitudes above 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 6Km up to 2Km. Accounting for the typical size of the plumes above Km shows the consistency of the Poisson statistics with the data. Low order conditional and joint statistics of M and are less compatible with the predictions of the Cohen and Craig (26). Stochastic variability of mass flux in a cloud resolving simulation p.43/43