Post on 01-Apr-2018
Development of a stochastic convectionscheme
R. J. Keane, R. S. Plant, N. E. Bowler, W. J. Tennant
Development of a stochastic convection scheme – p.1/45
Outline• Overview of stochastic parameterisation.• How the Plant Craig stochastic convective
parameterisation scheme works and the 3Didealised 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.3/45
Ensemble Forecasting & Stochastic Paramterisa-
tion• Single Deterministic Forecast:
E0(X, t) = A(E0,X, t) + P(E0);E0(X, 0) = I(X)
• Ensemble of Deterministic Forecasts:Ej(X, t) = A(Ej,X, t) + P(Ej);Ej(X, 0) = I(X) + Dj(X)
• Ensemble of Stochastic Forecasts:Ej(X, t) = A(Ej,X, t) + Pj(Ej, t);Ej(X, 0) = I(X) + Dj(X)
Development of a stochastic convection scheme – p.4/45
How stochastic parameterisations may improve ensemble
forecasts: noise-induced drift
Development of a stochastic convection scheme – p.5/45
How stochastic parameterisations may improve ensemble
forecasts: better forecast of variability
Development of a stochastic convection scheme – p.6/45
Heavy Rain Devon and SouthWales 6th June 2009Ken Mylne, 4th SRNWP workshop on ShortRange Ensemble Prediction Systems, 2009.
© Crown copyright Met Office 4th SRNWP Workshop on short-range EPS, Exeter June 2009.
Heavy rain Devon & S. Wales Sat 6th June 2009
MOGREPS0600 5th June
ECMWF0000 5th June
Development of a stochastic convection scheme – p.8/45
Climate modelling of heavy pre-cipitationSun et. al. J. Clim. 2006
Development of a stochastic convection scheme – p.9/45
Conventional convective param-eterisationFor a constant large-scale situation, aconventional parameterisaion models theconvection independently of space:
Development of a stochastic convection scheme – p.10/45
Conventional convective param-eterisationThis leads to a uniform, mean value ofconvection whatever the grid box size:
Development of a stochastic convection scheme – p.11/45
Stochastic parameterisationA stochastic scheme allows the number andstrength of clouds to vary consistent with thelarge-scale situation:
Development of a stochastic convection scheme – p.12/45
Effect of ParamterisationOf course, this has no effect if the grid box islarge enough:
Development of a stochastic convection scheme – p.13/45
Stochastic ParameterisationBut for a smaller gridbox ...
Development of a stochastic convection scheme – p.14/45
Effect of ParamterisationThe scheme allows some convective variability:
Development of a stochastic convection scheme – p.15/45
The real worldThe distribution will be different in reality, but thevariability will be similar.
Development of a stochastic convection scheme – p.18/45
Satellite image exampleNERC SRS, Dundee, 15/06/09 13:32.
Development of a stochastic convection scheme – p.19/45
Convection parameterisationschemes
• Trigger function• Mass-flux plume model• Closure• Examples
• Gregory Rowntree (UM standard)• Kain Fritsch• Plant Craig (based on Kain Fritsch)
Development of a stochastic convection scheme – p.22/45
Plant Craig scheme: Analogies
Statistical Mechanics Convection
Particle Cloud
Energy per particle Mass flux per cloud m
Number of particles Number of clouds N
Ensemble average energy Ensemble average mass flux 〈M〉
Temperature Ensemble mean mass flux per cloud 〈m〉
Entropy Ensemble mean number of clouds 〈N〉
Development of a stochastic convection scheme – p.23/45
Plant Craig scheme: Methodol-ogy
• Obtain the large-scale state by averagingresolved flow variables over both space andtime.
• Obtain 〈M〉 from CAPE closure and definethe equilibrium distribution of m (Cohen-Craigtheory).
• Draw randomly from this distribution to obtaincumulus properties in each grid box.
• Compute tendencies of grid-scale variablesfrom the cumulus properties.
Development of a stochastic convection scheme – p.24/45
Plant Craig scheme: ProbabilitydistributionAssuming a statistical equilibrium leads to anexponential distribution of mass fluxes per cloud:
p(m)dm =1
〈m〉exp
(
−m
〈m〉
)
dm.
So if m ∼ r2 then the probability of initiating aplume of radius r in a timestep dt is
〈M〉2r
〈m〉〈r2〉exp
(
−r2
〈r2〉
)
drdt
T.
Development of a stochastic convection scheme – p.26/45
Exponential distribution in aCRMCohen, PhD thesis (2001)
Development of a stochastic convection scheme – p.27/45
Ensemble Forecasting & Stochastic Paramterisa-
tion• Single Deterministic Forecast:
E0(X, t) = A(E0,X, t) + P(E0);E0(X, 0) = I(X)
• Ensemble of Deterministic Forecasts:Ej(X, t) = A(Ej,X, t) + P(Ej);Ej(X, 0) = I(X) + Dj(X)
• Ensemble of Stochastic Forecasts:Ej(X, t) = A(Ej,X, t) + Pj(Ej, t);Ej(X, 0) = I(X) + Dj(X)
Development of a stochastic convection scheme – p.28/45
PDF of total mass fluxAssuming that clouds are non-interacting, p(m)can be combined with a Poisson distribution forcloud number,
p(N) =〈N〉Ne−〈N〉
N !,
leading to the following distribution for total massflux:
p(M) =
(
〈N〉
〈m〉
)1/2
e−(〈N〉+M/〈m〉)M−1/2I1
(
2
√
〈N〉
〈m〉M
)
.
Development of a stochastic convection scheme – p.29/45
PDFs of mass flux in an SCM
• Plant & Craig, JAS, 2008
Development of a stochastic convection scheme – p.30/45
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.31/45
Rainfall snapshots: GregoryRowntree scheme
animationDevelopment of a stochastic convection scheme – p.32/45
Rainfall snapshots: Kain Fritschscheme
animationDevelopment of a stochastic convection scheme – p.33/45
Rainfall snapshots: Plant Craigscheme
animationDevelopment of a stochastic convection scheme – p.34/45
PDFs ofm andM for maximumaveragingAveraging area: 480 km square.Averaging time: 1 hour.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 108
−25
−24
−23
−22
−21
−20
−19
−18
−17
m (kg s−1)
ln (
p(m
) (k
g−
1s)
)
schemetheory
0 1 2 3 4 5 6
x 108
0
1
2
3
4
5
6x 10
−9
M (kg s−1)
p(M
) (k
g−
1 s
)
schemetheory
Development of a stochastic convection scheme – p.36/45
PDFs ofm andM for no averag-ing
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 108
−25
−24
−23
−22
−21
−20
−19
−18
−17
m (kg s−1)
ln (
p(m
) (k
g−
1s)
)
schemetheory
0 1 2 3 4 5 6 7 8
x 108
0
1
2
3
4
5
6
7x 10
−9
M (kg s−1)
p(M
) (k
g−
1 s
)
schemetheory
Development of a stochastic convection scheme – p.37/45
PDFs ofm andM for intermedi-ate averagingAveraging area: 160 km square.Averaging time: 1 hour.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 108
−26
−25
−24
−23
−22
−21
−20
−19
−18
−17
m (kg s−1)
ln (
p(m
) (k
g−
1s)
)
schemetheory
0 1 2 3 4 5 6
x 108
0
1
2
3
4
5
6
7x 10
−9
M (kg s−1)
p(M
) (k
g−
1 s
)
schemetheory
Development of a stochastic convection scheme – p.38/45
PDFs ofm andM for 16 km (noaveraging)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 108
−25
−24
−23
−22
−21
−20
−19
−18
−17
m (kg s−1)
ln (
p(m
) (k
g−
1s)
)
schemetheory
0 0.5 1 1.5 2 2.5 3 3.5 4
x 108
0
1
2
3
4
5
6
7x 10
−9
M (kg s−1)
p(M
) (k
g−
1 s
)
schemetheory
Development of a stochastic convection scheme – p.39/45
PDFs ofm andM for 16 km (in-termediate averaging)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 108
−25
−24
−23
−22
−21
−20
−19
−18
−17
m (kg s−1)
ln (
p(m
) (k
g−
1s)
)
schemetheory
0 0.5 1 1.5 2 2.5 3 3.5 4
x 108
0
1
2
3
4
5
6
7x 10
−9
M (kg s−1)
p(M
) (k
g−
1 s
)
schemetheory
Development of a stochastic convection scheme – p.40/45
Case study: CSIP IOP18• Starts at 25th August 2005, 07:00.• 12 km grid with 146 × 182 grid points.
20 40 60 80 100 120
20
40
60
80
100
120
140
160
180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
Development of a stochastic convection scheme – p.41/45
Ensemble of 6 runs using PCscheme
20 40 60 80100120
50
100
150
20 40 60 80100120
50
100
150
20 40 60 80100120
50
100
150
20 40 60 80100120
50
100
150
20 40 60 80100120
50
100
150
20 40 60 80100120
50
100
150
Development of a stochastic convection scheme – p.42/45
Rainfall against time for each ofthree schemes
8 10 12 14 16 18 200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
mea
n ra
infa
ll (kg
m−2
)
time (hours)
PC cvPC lsPC totGR cvGR lsGR totKF cvKF lsKF tot
Development of a stochastic convection scheme – p.43/45
Future 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 theensemble and the spread-error relationship.
Development of a stochastic convection scheme – p.44/45
Conclusions• The convective variability in the scheme is according to
the Cohen Craig theory, and is not due to spuriousnoise from the large-scale.
• An averaging area of roughly 160 km is required toeffect this.
• The statistical behaviour of the scheme is correct atdifferent resolutions, although the amount of averagingrequired may vary.
• The scheme behaves sensibly in a mesoscale setup,and is ready to be implemented in an ensembleprediction system.
Development of a stochastic convection scheme – p.45/45