Development of a stochastic convection scheme R. J. Keane, R. S. Plant, N. E. Bowler, W. J. Tennant Development of a stochastic convection scheme p.1/44
Outline Overview of stochastic parameterisation. How the Plant Craig stochastic convective parameterisation scheme works and the 3D idealised setup. Results: rainfall statistics. A look at the Plant Craig scheme in a mesoscale run. Conclusions and future work. Development of a stochastic convection scheme p.2/44
Ensemble Forecasting & Stochastic Paramterisation Single Deterministic Forecast: Ė 0 (X,t) = A(E 0,X,t) + P(E 0 ); E 0 (X, 0) = I(X) Ensemble of Deterministic Forecasts: Ė j (X,t) = A(E j,x,t) + P(E j ); E j (X, 0) = I(X) + D j (X) Ensemble of Stochastic Forecasts: Ė j (X,t) = A(E j,x,t) + P j (E j,t); E j (X, 0) = I(X) + D j (X) Development of a stochastic convection scheme p.3/44
How stochastic parameterisations may improve ensemble forecasts: better forecast of variability Development of a stochastic convection scheme p.4/44
Rank Histograms of surface temperature Development of a stochastic convection scheme p.5/44
Conventional convective parameterisation For a constant large-scale situation, a conventional parameterisaion models the convection independently of space: Development of a stochastic convection scheme p.6/44
Conventional convective parameterisation This leads to a uniform, mean value of convection whatever the grid box size: Development of a stochastic convection scheme p.7/44
Stochastic parameterisation A stochastic scheme allows the number and strength of clouds to vary consistent with the large-scale situation: Development of a stochastic convection scheme p.8/44
Effect of Paramterisation Of course, this has no effect if the grid box is large enough: Development of a stochastic convection scheme p.9/44
Stochastic Parameterisation But for a smaller gridbox... Development of a stochastic convection scheme p.10/44
Effect of Paramterisation The scheme allows some convective variability: Development of a stochastic convection scheme p.11/44
Stochastic Parameterisation Development of a stochastic convection scheme p.12/44
Effect of Paramterisation Development of a stochastic convection scheme p.13/44
The real world The distribution will be different in reality, but the variability will be similar. Development of a stochastic convection scheme p.14/44
Scale separation: thermodynamics Development of a stochastic convection scheme p.15/44
Scale separation: rainfall Development of a stochastic convection scheme p.16/44
Convection schemes Trigger function Mass-flux plume model Closure Examples parameterisation Gregory Rowntree (UM standard) Kain Fritsch Plant Craig (based on Kain Fritsch) Development of a stochastic convection scheme p.17/44
Plant Craig scheme: Methodology Obtain the large-scale state by averaging resolved flow variables over both space and time. Obtain M from CAPE closure and define the equilibrium distribution of m (Cohen-Craig theory). Draw randomly from this distribution to obtain cumulus properties in each grid box. Compute tendencies of grid-scale variables from the cumulus properties. Development of a stochastic convection scheme p.18/44
Plant Craig scheme: Averaging area Development of a stochastic convection scheme p.19/44
Plant Craig scheme: Probability distribution Assuming a statistical equilibrium leads to an exponential distribution of mass fluxes per cloud: p(m)dm = 1 ( ) m m exp dm. m So if m r 2 then the probability of initiating a plume of radius r in a timestep dt is ( ) M 2r r 2 m r 2 exp dr dt r 2 T. Development of a stochastic convection scheme p.20/44
Exponential distribution in a CRM Cohen, PhD thesis (2001) Development of a stochastic convection scheme p.21/44
PDF of total mass flux Assuming that clouds are non-interacting, p(m) can be combined with a Poisson distribution for cloud number, p(n) = N N e N N! leading to the following distribution for total mass flux: ( ) ( ) 1/2 N p(m) = e ( N +M/ m ) M 1/2 N I 1 2 m m M., Development of a stochastic convection scheme p.22/44
PDFs of mass flux in an SCM Plant & Craig, JAS, 2008 Development of a stochastic convection scheme p.23/44
3D Idealised UM setup Radiation is represented by a uniform cooling. Convection, large scale precipitation and the boundary layer are parameterised. The domain is square, with bicyclic boundary conditions. The surface is flat and entirely ocean, with a constant surface temperature imposed. Targeted diffusion of moisture is applied. The grid size is 32 km. Development of a stochastic convection scheme p.24/44
Rainfall snapshots: Gregory Rowntree scheme animation Development of a stochastic convection scheme p.25/44
Rainfall snapshots: Kain Fritsch scheme animation Development of a stochastic convection scheme p.26/44
Rainfall snapshots: Plant Craig scheme animation Development of a stochastic convection scheme p.27/44
Model grid division 16km 32km Development of a stochastic convection scheme p.28/44
PDFs of m and M for maximum averaging Averaging area: 480 km square. Averaging time: 50 minutes. 17 18 scheme theory 6 x 10 9 5 scheme theory 19 ln ( p(m) (kg 1 s) ) 20 21 22 23 p(m) (kg 1 s) 4 3 2 24 1 25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m (kg s 1 ) x 10 8 0 0 1 2 3 M (kg s 1 ) 4 5 6 x 10 8 Development of a stochastic convection scheme p.29/44
PDFs of m and M for no averaging 17 18 scheme theory 7 x 10 9 6 scheme theory 19 5 ln ( p(m) (kg 1 s) ) 20 21 22 p(m) (kg 1 s) 4 3 23 2 24 1 25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0 1 2 3 4 5 6 7 8 m (kg s 1 ) x 10 8 M (kg s 1 ) x 10 8 Development of a stochastic convection scheme p.30/44
PDFs of m and M for intermediate averaging Averaging area: 160 km square. Averaging time: 50 minutes. 17 18 scheme theory 7 x 10 9 6 scheme theory 19 5 20 ln ( p(m) (kg 1 s) ) 21 22 p(m) (kg 1 s) 4 3 23 2 24 25 1 26 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0 1 2 3 4 5 6 m (kg s 1 ) x 10 8 M (kg s 1 ) x 10 8 Development of a stochastic convection scheme p.31/44
PDFs of m and M for less averaging Averaging area: 96 km square. Averaging time: 50 minutes. 17 scheme theory 7 x 10 9 scheme theory 18 6 19 5 ln ( p(m) (kg 1 s) ) 20 21 22 p(m) (kg 1 s) 4 3 23 2 24 1 25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m (kg s 1 ) x 10 8 0 0 1 2 3 4 5 6 Development M (kg s 1 ) of a stochastic convection scheme x 10 8 p.32/44
PDFs of m and M for 10 K/day cooling Averaging area: 160 km square. Averaging time: 40 minutes. 17 scheme theory 7 x 10 9 scheme theory 18 6 19 5 ln ( p(m) (kg 1 s) ) 20 21 22 p(m) (kg 1 s) 4 3 23 2 24 1 25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m (kg s 1 ) x 10 8 0 0 1 2 3 4 5 6 Development M (kg s 1 ) of a stochastic convection scheme x 10 8 p.33/44
PDFs of m and M for 12 K/day cooling Averaging area: 160 km square. Averaging time: 33 minutes. 17 scheme theory 6 x 10 9 scheme theory 18 5 19 ln ( p(m) (kg 1 s) ) 20 21 22 p(m) (kg 1 s) 4 3 2 23 24 1 25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m (kg s 1 ) x 10 8 0 0 1 2 3 4 5 6 Development M (kg s 1 ) of a stochastic convection scheme x 10 8 p.34/44
PDFs of m and M for 16 km Averaging area: 144 km square. Averaging time: 67 minutes. 17 18 scheme theory 7 x 10 9 6 scheme theory 19 5 ln ( p(m) (kg 1 s) ) 20 21 22 p(m) (kg 1 s) 4 3 23 2 24 1 25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 m (kg s 1 ) x 10 8 M (kg s 1 ) x 10 8 Development of a stochastic convection scheme p.35/44
PDFs of m and M for 51 km Averaging area: 152 km square. Averaging time: 51 minutes. 17 18 scheme theory 4.5 x 10 9 4 scheme theory 19 3.5 ln ( p(m) (kg 1 s) ) 20 21 22 p(m) (kg 1 s) 3 2.5 2 1.5 23 1 24 0.5 25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m (kg s 1 ) x 10 8 0 0 1 2 3 4 5 6 7 8 Development M (kg s 1 ) of a stochastic convection scheme x 10 8 p.36/44
Case study: CSIP IOP18 Starts at 25th August 2005, 07:00. 12 km grid with 146 182 grid points. 180 160 140 120 100 80 60 40 20 x 10 3 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 20 40 60 80 100 120 0 Development of a stochastic convection scheme p.37/44
Comparison between models and rainfall radar: 1000Z Gregory Rowntree Kain Fritsch Multiplicative Noise Plant Craig Development of a stochastic convection scheme p.38/44
Comparison between models and rainfall radar: 1400Z Gregory Rowntree Kain Fritsch Multiplicative Noise Plant Craig Development of a stochastic convection scheme p.39/44
Average rainfall against time for different models 1 x 10 4 Convective rain 1 x 10 4 Large scale rain 1 x 10 4 Total rain 0.9 0.9 0.9 0.8 0.8 0.8 mean rainfall (kg m 2 s 1 ) 0.7 0.6 0.5 0.4 0.3 mean rainfall (kg m 2 s 1 ) 0.7 0.6 0.5 0.4 0.3 mean rainfall (kg m 2 s 1 ) 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 8 10 12 14 16 18 20 time (hours) 0 8 10 12 14 16 18 20 time (hours) 0 8 10 12 14 16 18 20 time (hours) Development of a stochastic convection scheme p.40/44
RMS of rainfall against time for two stochastic models red Plant Craig; blue Multiplicative Noise full line RMS convection; dotted line mean convection 1.2 x 10 4 1 rainfall (kg m 2 s 1 ) 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 12 14 time since start of run (hours) Development of a stochastic convection scheme p.41/44
Current work Implement the PC scheme in MOGREPS, to determine its impact on variability. Run on NAE domain ( 20 km), for one Summer month. Compare with existing GR run and deterministic version of PC. Look at the effect of the scheme, and its stochastic nature, on the variability of the ensemble and the spread-error relationship. Development of a stochastic convection scheme p.42/44
Possible future work Select extreme events from a data set and investigate whether or not the characterisation of the high rainfall tail is improved by including the Plant Craig scheme. Run convection schemes offline, driven with coarse-grained dynamics, and investigate their characterisation of the variability of the rainfall. Development of a stochastic convection scheme p.43/44
Conclusions The convective variability in the scheme is according to the Cohen Craig theory, and is not due to spurious noise from the large-scale. An averaging area of roughly 160 km is required to effect this. The scheme behaves sensibly in a mesoscale setup, and is ready to be implemented in an ensemble prediction system. Some work needs to be done to increase the convective fraction of the rainfall when using the scheme. Development of a stochastic convection scheme p.44/44