Stochastic Representation of Convectionsws00rsp/research/stochastic/RMStalk.pdf · random subgrid variability. ... Stochastic Representation of Convection ... (deterministic limit

Stochastic Representation ofConvection

RMS Dynamical Problems Group9th June 2005

Bob Plant and George Craigr.s.plant@rdg.ac.uk

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