Mass flux fluctuation in a cloud resolving simulation with diurnal forcing

Transcription

1 Mass flux fluctuation in a cloud resolving simulation with diurnal forcing Jahanshah Davoudi Norman McFarlane, Thomas Birner Physics department, University of Toronto Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p./46

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, Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.2/46

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. Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.3/46

4 Parameterization Basics Arakawa & Schubert 974 Key Quasi-equilibrium assumption: τ adj τ ls Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.4/46

5 Quasi-equilibrium Convective equilibrium requires scale separation Large scale uniform over region containing many clouds Large scale slowly varying so convection has time to respond Convective activity within a small grid cell is highly variable(even in statistically stationary state) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.5/46

6 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? Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.6/46

7 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 Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.7/46

8 Other constraints 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. Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.8/46

9 mass flux of individual clouds are statistically un-correlated : given λ = /( m ) = N M P M (n) = Prob{N [(, 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 N Poisson distributed plumes is a Compound point process: M = NX i= m i Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.9/46

10 Predicted distribution So the Generating function of M is calculated exactly: G(t) = e tm m,n = e t P N i m i m,n e tm m,n = g N (t) N = e Λ e Λg(t) where g(t) = e tm m Λ = N Therefore the probability distribution of the total mass flux is exactly given by: P(M) = P(M) = «N /2 e N M /2 e M/ m I 2 m «! N /2 m M All the moments of M are analytically tractable and are functions of N and m. (δm) 2 M 2 = 2 N (δm) 3 M 3 = 6 N 2 Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p./46

11 Estimates In a region with area A and grid size x L where L the mean cloud spacing is: L = (A/ N ) /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 Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p./46

12 CRM distributions of cloud mass flux Mass flux per cloud Total mass flux distribution Craig and Cohen JAS (26) Resolution: Domain: Boundary conditions: Forcings: 2km 2km 5 levels 28 km 28 km 2 km doubly periodic, fixed SST of 3 K fixed tropospheric cooling of 2,4,8,2,6 K day Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.2/46

13 CRM mass flux variance Without organization Shear (organization) Left panel: normalized standard deviation of are-integrated convective mass flux versus characteristic cloud spacing. Right panel: Various degrees of convection organization: un-sheared(*), weak shear ( ), strong shear(+). Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.3/46

14 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 Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.4/46

15 A 2D cut through the convective field Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.5/46

16 In cloud properties up-drafts M(t) and A(t) M(t) (x Kg /sec).2 z=7.34 m A(t) 6 z=7.34 m t(days) t(days).4.2 m c (t)(x Kg /sec) t(days) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.6/46

17 Long time portraits of ql (t), qv (t) and T(t) 25 2 z=57 m z=57 m qv (x -6 kg/kg) t (day) t (day) z=57 m T ( K) ql (x -6 kg/kg) t (day) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.7/46

18 short time portraits of q l (t), q v (t) and T(t) q l (x -6 kg/kg) 2 8 z=57 m t (day) q v (x -6 kg/kg) z=57 m t (day) z=57 m 29.7 T ( K) t (day) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.8/46

19 Distribution of CAPE and LNB 3 25 CAPE (J Kg - ) time (day) CAPE p CAPE r P(CAPE) P(LNB) CAPE (J Kg - ) LNB (m) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.9/46

20 Short and long wave heating rates t (day) M(t) (x 8 Kg s - ) Q L (K/day) Q S (K/day) z (Km) z (Km) < Q L > (K/day) < Q S > (K/day) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.2/46

21 Adjustment time.8.6 z=57 m z= m z=843.6 m z=4. m C QT M (τ) τ (hours) C QT M(τ) = Q T (t) M(t + τ) σ QT σ M The adjustment time of the total heating rate Q T = Q S + Q L and the mass flux at various altitudes. The τ adj varies in the range of 2 4 hours. Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.2/46

22 Auto-correlation of up-drafts M C M (,τ) τ (hours) z=56.9 m z= z=99 m z=29 m C M (,τ) τ (hours) z=56.9 m z= z=99 m z=29 m C M (δ, τ) = M(z, t) M(z + δ, t + τ) σ M (z)σ M (z + δ) where M M M and is a time average. Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.22/46

23 Auto-correlation of up-drafts A C A (,τ) τ (hours) z=56.9 m z= z=99 m z=29 m C A (,τ) τ (hours) z=56.9 m z= z=99 m z=29 m Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.23/46

24 Auto-correlation of up-drafts m c.8 z=56.9 m z= z=99 m z=29 m.8 z=56.9 m z= z=99 m z=29 m.6.6 C mc (,τ).4 C mc (,τ) τ (hours) τ (hours) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.24/46

25 Auto-correlation of q v z=99 m.5.95 z=57 m z= m z=843 m C qv (δ,τ) C qv (,τ) τ (day) τ (day) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.25/46

26 Auto-correlation of q l.8 z=843 m.8 z=57 m z= m z=843 m.6.6 C ql (,τ).4.2 C ql (,τ) τ (day) τ (day) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.26/46

27 Auto-correlation of T z=843 m.2.8 z=57 m z=843 m z=4 m C T (,τ).4.2 C T (,τ) τ (day) τ (day) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.27/46

28 CAPE auto-correlation C cape (,τ).4.2 C cape (,τ) τ (days) τ (days) Speculation: τ rad = H2 ρc p (dθ/dz) Q rad Assuming Q rad = 25Wm 2, H = 5Km, ρ =.6kgm 3, dθ/dz = 3Kkm gives w s.5cms and τ rad 3 days. Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.28/46

29 Spatio-temporal delayed correlation function C M (δ,τ) τ (hours) δ= m δ=84.2 m δ= m δ= m δ= m δ= m z= m δ(m) w * =5.5 m/s δ= w * τ max τ max ( hours) z (km) <w c > (m/s) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.29/46

30 Information transport in A u < A(z,t) A(z+δ,t+τ) > δ= m δ=522.5 m δ=24.5 m δ=89.3 m δ= m δ= m z=56 m < A(z,t) A(z+δ,t+τ) > δ= m δ=84.2 m δ= m δ= m δ= m δ= m z= m τ (hours) τ (hours) < A(z,t) A(z+δ,t+τ) > τ (hours) δ= m δ=597.6 m δ=96.85 m δ= m δ= m δ= m z=843.6 m Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.3/46

31 Information transport in M u < M(z,t) M(z+δ,t+τ) > δ= m δ=522.5 m δ=24.5 m δ=89.3 m δ= m δ= m z=56 m < M(z,t) M(z+δ,t+τ) > δ= m δ=84.2 m δ= m δ= m δ= m δ= m z= m τ (hours) τ (hours) < M(z,t) M(z+δ,t+τ) > τ (hours) δ= m δ=597.6 m δ=96.85 m δ= m δ= m δ= m z=843.6 m Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.3/46

32 Information transport in W u < w(z,t) w(z+δ,t+τ) > δ= m δ=522.5 m δ=24.5 m δ=89.3 m δ= m δ= m z=56 m < w(z,t) w(z+δ,t+τ) > δ= m δ=84.2 m δ= m δ= m δ= m δ= m z= m τ (hours) τ (hours) < w(z,t) w(z+δ,t+τ) > δ= m δ=597.6 m δ=96.85 m δ= m δ= m δ= m z=843.6 m τ (hours) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.32/46

33 Mean characteristics z (Km) 5 z (Km) <M> (Kg m -2 s - ) <A> Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.33/46

34 N and σ c z (Km) 8 6 z (Km) <N> <σ c > (x 4x 6 m 2 ) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.34/46

35 Fat tails of marginal PDF of up-draft area coverage P(A;z)... z=56 m z=264 m z= m P(A;z)... z= m z=583 m z=729 m z=843 m z=96 m z=8 m e-4 e-4 e e A A. z=8 m z=2 m z=32 m z=44 m z=56 m P(A;z).. e-4 e A Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.35/46

36 Altitude variability of total Mass flux PDF z = 57 m z = m 6 4 Measured P(M) at z=57 m Fit with free M and N Prediction with measured M and N 4 2 Measured P(M) at z=2976 m Fit with free M and N Prediction with measured M and N 2 8 P(M;z) 8 P(M;z) M (x Kg s ) M (x Kg s ) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.36/46

37 P(M,z) z = m z = m 4 2 Measured P(M) at z= m Fit with free M and N Prediction with measured M and N 25 Measured P(M) at z= m Fit with free M and N Prediction with measured M and N P(M;z) P(M;z) M (x Kg s ) M (x Kg s ) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.37/46

38 P(M,z) z = 87.3 m z = 99. m 25 Measured P(M) at z=843.6 m Fit with free M and N Prediction with measured M and N 35 3 Measured P(M) at z=99 m Fit with free M and N Prediction with measured M and N P(M;z) P(M;z) M (x Kg s ) M (x Kg s ) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.38/46

39 P(M,z) z = 4. m z = 29. m 5 45 Measured P(M) at z=4 m Fit with free M and N Prediction with measured M and N 7 6 Measured P(M) at z=29 m Fit with free M and N Prediction with measured M and N P(M;z) 25 P(M;z) M (x Kg s ) M (x Kg s ) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.39/46

40 Mass flux variance 4 2 <N><(δM) 2 >/<M> 2 z (Km) 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 N Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.4/46

41 Mass flux Skewness 4 2 <N> 2 <(δm) 3 >/<M> 3 z (Km) The prediction: (δm) 3 M 3 = 6 N 2 Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.4/46

42 Clustering degree If N(A) is the number of clouds in any sub-area A S then P p [N(A) = k] is defined by P p [N(A) = k] = (γ A )k e γ A k! for k =,,, where γ A is the average number of clouds in the sub-area with size A. Centered on any arbitrary cloud we define the probability of finding the farthest neighboring cloud with a given Euclidean distance less than r, i.e. Π < p (r). Π < p (r) = Π> p (r) = P p(n(a) = ) = e γπr2, where A = πr 2 is used. For obtaining the clustering degree one measures directly the cumulative probability of having k clouds inside a ball of radius r centered around any existing cloud in each altitude. Then χ(r) = Π< (r) Π < p (r). Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.42/46

43 Clustering I Π < (r)/π < p (r) r (Km) z=233 m In a radius of around km any cloud is surrounded with more neighboring clouds than a Poisson distribution predicts. Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.43/46

44 Clustering II m <N>=5.288 <N>= z= m <N>=3.58 <N>=3.535 P(N,z) P(N,z) N N 4 35 σ Ν 2 <N> 3 z (Km) Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.44/46

45 Summary Analysis of the time scales shows that a state of quasi-equilibrium establishes in our CRM simulation with diurnal forcing. The response of the total up-draft mass flux to the total heating rate at all heights indicates to a range of 2 4 hours adjusting time. The statistics of the total up-draft mass flux is qualitatively consistent with the predictions of the Cohen and Craig (26). The Gibbs theory under-estimates the variance and skewness of the total mass flux. Analysis of our CRM simulation shows that the non-interacting assumption employed in the Craig and Cohen theory does not hold as we demonstrate the clouds preferentially cluster. Mass flux fluctuation in a cloud resolving simulation with diurnal forcing p.45/46