Stochastic Representation of Convection RMS Dynamical Problems Group 9th June 2005 Bob Plant and George Craig [email protected] Department of Meteorology, University of Reading
Stochastic Representation ofConvection
RMS Dynamical Problems Group9th June 2005
Bob Plant and George [email protected]
Department of Meteorology,
University of Reading
Outline
1. The need for a stochastic representation of convection
2. Some experiments so far
3. A stochastic scheme
4. Tests of scheme
5. Outlook
Stochastic Representation of Convection – p.1/30
Why a stochastic representation?
Stochastic Representation of Convection – p.2/30
A much harder question is...
What makes you think you can getaway with using a deterministic
representation?
OStochastic Representation of Convection – p.3/30
A much harder question is...
What makes you think you can getaway with using a deterministic
representation?
Stochastic Representation of Convection – p.3/30
Argument for Stochastic Approach
1. A deterministic scheme gives unique increments due toconvection for a given large-scale state
2. A major source of variability is that convective instability isreleased in discrete events
3. The number of events in a GCM grid-box is not largeenough to produce a steady response to a steady forcing
4. Wide range of sub-grid states are possible, so aim tocalculate their ensemble mean effect
Fluctuating component of sub-grid motions may have importantinteractions with large-scale
OStochastic Representation of Convection – p.4/30
Argument for Stochastic Approach
1. A deterministic scheme gives unique increments due toconvection for a given large-scale state
2. A major source of variability is that convective instability isreleased in discrete events
3. The number of events in a GCM grid-box is not largeenough to produce a steady response to a steady forcing
4. Wide range of sub-grid states are possible, so aim tocalculate their ensemble mean effect
Fluctuating component of sub-grid motions may have importantinteractions with large-scale
OStochastic Representation of Convection – p.4/30
Argument for Stochastic Approach
1. A deterministic scheme gives unique increments due toconvection for a given large-scale state
2. A major source of variability is that convective instability isreleased in discrete events
3. The number of events in a GCM grid-box is not largeenough to produce a steady response to a steady forcing
4. Wide range of sub-grid states are possible, so aim tocalculate their ensemble mean effect
Fluctuating component of sub-grid motions may have importantinteractions with large-scale
OStochastic Representation of Convection – p.4/30
Argument for Stochastic Approach
1. A deterministic scheme gives unique increments due toconvection for a given large-scale state
2. A major source of variability is that convective instability isreleased in discrete events
3. The number of events in a GCM grid-box is not largeenough to produce a steady response to a steady forcing
4. Wide range of sub-grid states are possible, so aim tocalculate their ensemble mean effect
Fluctuating component of sub-grid motions may have importantinteractions with large-scale
OStochastic Representation of Convection – p.4/30
Argument for Stochastic Approach
1. A deterministic scheme gives unique increments due toconvection for a given large-scale state
2. A major source of variability is that convective instability isreleased in discrete events
3. The number of events in a GCM grid-box is not largeenough to produce a steady response to a steady forcing
4. Wide range of sub-grid states are possible, so aim tocalculate their ensemble mean effect
Fluctuating component of sub-grid motions may have importantinteractions with large-scale
Stochastic Representation of Convection – p.4/30
Range of States
0
10
20
30
40
50
60
70
0 0.05 0.1 0.15 0.2 0.25 0.3
Freq
uenc
y (sc
aled)
Total mass flux at 2km (kg/s)
16km
32km
64km
Total mass flux (kg/s) 0.30
Distribution of massfluxes in CRM simulationof radiative-convectiveequilibrium over ocean.Uniform SST and forcedwith constant tropo-spheric cooling. Averagedover various areas.
Also Xu et al (1992);Shutts and Palmer (2004)
Stochastic Representation of Convection – p.5/30
Practical Motivations
Stochastic parameterizations may resolve known problems withcurrent approaches:
NWP models have insufficient ensemble spread
Buizza et al (2005)
OStochastic Representation of Convection – p.6/30
Practical Motivations
Stochastic parameterizations may resolve known problems withcurrent approaches:
NWP models have insufficient ensemble spread
Buizza et al (2005) Stochastic Representation of Convection – p.6/30
Practical Motivations
Stochastic parameterizations may resolve known problems withcurrent approaches:
NWP models have insufficient ensemble spread(improvement expected)
Low frequency variability (improvements likely)Marginal predictability of some events which react strongly to
near-grid-scale noise (Zhang et al 2003)
GCMs have insufficient variability in tropics (impact on QBO)
Systematic model errors (hopeful of improvements)eg, propagation of convection
Stochastic Representation of Convection – p.7/30
Not a magic wand - some problemswill not go away
Stochastic Representation of Convection – p.8/30
Existing Variability
Existing parameterizations do have variability, but it is:
unphysical (numerical)
uncontrolled
does not exhibit the correct dependencies
Stochastic Representation of Convection – p.9/30
Example of Artificial Variability
0
0.5
1
1.5
2
2.5
3
50 100 150 200 250
Flux/m
ean f
lux
Timestep number
Kain−Fritsch
Normalized response to aconstant forcing by Kain-Fritsch scheme over oneday in a SCM
No dependence on (for example) grid size.
Stochastic Representation of Convection – p.10/30
Some stochastic experiments
Stochastic Representation of Convection – p.11/30
Variability in Model Formulation
In ECMWF ensemble system, scale parameterizationtendencies,
Tendency = D+(1+ ε)P
Improves ensemble spread
Bright and Mullen (2002): stochastic perturbation to KFtrigger.Increased skill and dispersion of short-range precipitation forecasts
Lin and Neelin (2002): add noise to CAPE closure ofZhang/Macfarlane scheme in CCM3.Increase variance of daily tropical precipitation
Khouider, Majda and Katsoulakis (2003). Spin-flip model.Sites within each grid box that may or may not support deep convection.
Convective heating scales with fractional area. Stochastic Representation of Convection – p.12/30
Aim
To construct a stochastic scheme in which
the character and strength of the noise has a physicalbasis
the physical basis is supported (or inspired) by CRMstudies
physical noise >> numerical noise from scheme
noise → 0 if there are very many clouds and in this limitscheme behaves no worse than standard deterministicschemes
Stochastic Representation of Convection – p.13/30
A Stochastic Scheme
Stochastic Representation of Convection – p.14/30
Basic Structure
Mass-flux formalism (based on Kain-Fritsch)...
No trigger function. Presence of convection dictated byrandom subgrid variability.
Spectrum of possible plumes chosen from distribution ofmass fluxes. Each plume represents cloud of given massflux.
Clouds persist for finite lifetime 6= timestep.
CAPE closure to remove instability on a timescale thatdepends on forcing. Calculations performed on anaveraged (non-local) sounding.
Stochastic Representation of Convection – p.15/30
Statistical Mechanics I
Craig and Cohen (2004)
Weakly-interacting, point-like convective cells inequilibrium with large scale forcing have exponentialdistribution of mass flux per cloud
p(m)dm =1〈m〉
exp
(
−m〈m〉
)
dm
cf Boltzmann distribution of energies
Ensemble mean mass flux 〈M〉 and is mean mass flux percloud 〈m〉 functions of large-scale forcing only
Stochastic Representation of Convection – p.16/30
Example Distributions
10
100
0 1 2 3 4 5 6 7 8
Num
ber o
f clo
uds
Mass flux (x 10^7 kgm^2s^−1)
Distribution at 3.1km8K/d forcing
10
100
0 1 2 3 4 5 6 7 8
Num
ber o
f clo
uds
Mass flux (x 10^7 kgm^2s^−1)
Distribution at 1.3km16K/d forcing
Stochastic Representation of Convection – p.17/30
Statistical Mechanics II
Number of clouds in given region given by Poissondistribution if clouds randomly distributed in space.
This gives pdf of the total mass flux
p(M)=1
〈M〉
√
〈M〉
Mexp
(
−M + 〈M〉
〈m〉
)
I1
(
2〈m〉
√
〈M〉M
)
Deviations modest if a wind shear imposed
Stochastic Representation of Convection – p.18/30
〈m〉 ∼ constant at fixed level
0
2000
4000
6000
8000
10000
0 0.5 1 1.5 2 2.5 3
Heig
ht (m
)
Mean mass flux (x10^7 kg m^2 s^−1)
8K/d
0km
4K/d
10kmIncreased forcing pre-dominantly affects cloudnumber 〈N〉 = 〈M〉/〈m〉
not the mean w(scalings ofEmanuel and Bister1996; Grant andBrown 1999)
nor the mean size(Robe and Emanuel1996; Cohen 2001)
Stochastic Representation of Convection – p.19/30
Implications for Parameterization
In each grid box, probability of finding cloud of given mfrom exponential
〈m〉 taken as constant from CRM data
Behaviour of each cloud modelled based on 1DKain-Fritsch plume model
Exponential distribution imposed at LCL but distributionfree to evolve at other levels
Need closure for 〈M〉
Stochastic Representation of Convection – p.20/30
Closure I
CAPE closure based on full ensemble of clouds
CAPE removed with a closure timescale that varies withforcing
τ = k〈cloud separation〉 = kδx
√
〈m〉
〈M〉
Tolerant of weak forcing
Acts aggressively to remove large instability
Stochastic Representation of Convection – p.21/30
Adjustment Timescale
Closure timescaleequivalent toadjustmenttimescale if forcingremoved
Rapid responsegoverned by gravitywave propagationbetween clouds
(Slower evolution ofmoisture variables)
Time scaled by cloud separation
Stochastic Representation of Convection – p.22/30
Closure II
〈M〉 depends only the large-scale state
Local calculations appropriate only if no sub-gridfluctuations
Leads to amplification of any artifical local fluctuationsin deterministic mass flux scheme
Averaging region should contain many clouds
Stochastic Representation of Convection – p.23/30
SCM Tests
Stochastic Representation of Convection – p.24/30
Tests of scheme
Met Office Unified Model – single column version
parameterizations for boundary layer transport, stratiformcloud
forced as in CRM simulations (fixed tropospheric cooling)
CAPE closure based on sounding averaged over 100timesteps
Aim is to replicate mean state and fluctuations of a companionCRM simulation
Stochastic Representation of Convection – p.25/30
Physical not Numerical Noise
Does a steady forcing give a steady response (deterministiclimit of a large grid box)?
0
0.5
1
1.5
2
2.5
3
50 100 150 200 250
Flu
x/m
ean
flux
Timestep number
Kain−Fritsch
0
0.5
1
1.5
2
2.5
3
50 100 150 200 250
Flu
x/m
ean
flux
Timestep number
400km box
0
0.5
1
1.5
2
2.5
3
50 100 150 200 250F
lux/
mea
n flu
xTimestep number
64km box
1 ’cloud’ ∼ 200 clouds ∼ 5 clouds
Stochastic Representation of Convection – p.26/30
Distribution of M
Is the desired distribution of M obtained for finite-sized gridboxes?
0
20
40
60
80
100
120
0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08
Prob
abilit
y (sc
aled)
Total mass flux (kg/s)
0.04 0.08
400km
256km
0
5
10
15
20
25
30
35
0 0.02 0.04 0.06 0.08 0.1 0.12
Prob
abilit
y (sc
aled)
Total mass flux (kg/s)
128km
64km
0 0.12
Stochastic Representation of Convection – p.27/30
Realistic Mean State
Mean state temperature and humidity profiles sensible (notworse than Kain-Fritsch)?
Differences between SCM states and the CRM state arecomparable to differences between CRMs.
Fluctuations do not shift mean state (shouldn’t in 1D!)
Stochastic Representation of Convection – p.28/30
Cloud Properties
Are properties of the individual clouds sensible?
〈m〉 ∼ constant with height, exponential distribution?
0
2000
4000
6000
8000
10000
0 1 2 3 4 5 6 7 8
Heigh
t (m
)
Mean mass flux (x10^7 kg m^2 s^−1)
Stochastic scheme
10km
0km
CRM
10
100
1000
0 5 10 15 20
Numb
er of
clou
ds
Mass flux (x 10^7 kgm^2s^−1)
Stochastic schemeDistribution at 5.75km
Stochastic Representation of Convection – p.29/30
Future Steps
1. Implementation in full UM (non-trivial as non-local)
2. Implementation in DWD Lokal Model (regional NWPmodel)
3. Tests in COSMO-LEPS ensemble system, to includecases from CSIP
4. Dependencies of cloud lifetime (size and forcing) fromtracking experiments in CRMs
5. Relax (or remove) equilibrium assumption?(with Laura Davies and Steve Derbyshire)
6. Longer term ensemble tests
7. Aqua-planet global UM
Stochastic Representation of Convection – p.30/30