1Dry Convection

Phenomenology, Simulation and Parameterization of Atmospheric Convection

Pier Siebesma

Yesterday: “Dry” Atmospheric Convection

Today: “Moist” Convection and (shallow cumulus) clouds

2Dry Convection

o0

Equator

N30

Northoregion

windTrade

Subsidence

~0.5 cm/s

10 m/s

vE vE

inversion

Cloud base

~500m

Tropopause 10km

•Stratocumulus

•Interaction with radiation

•Shallow Convective Clouds

•Little precipitation

•Vertical turbulent transport

•No net latent heat production

•Fuel Supply Hadley Circulation

•Deep Convective Clouds

•Precipitation

•Vertical turbulent transport

• Net latent heat production

•Engine Hadley Circulation

EUROCS intercomparison project on cloud representation in GCM’s in the Eastern Pacific

Large Scale Models tend to overestimate Tradewind cumulus cloudiness and underestimate Stratocumulus

Siebesma et al. (2005, QJRMS)

scuShallow cuDeep cu

5Dry Convection

rlii

ll

l

vii

vv

v

radp

ii

Pecquxz

qwq

t

q

ecquxz

qwq

t

q

Qecc

Lu

xzw

t

v

v

v

Large scale Large scale

advectionadvection

Large scale Large scale

subsidencesubsidence

turbulent turbulent transporttransport

Net Net Condensation Condensation RateRate

Grid Averaged Equations of thermodynamic variables

Introduce moist conserved variables!

lp

l qcL

•Liquid water potential Temperature

lvt qqq •Total water specific humidity

rtt

tt

radll

ll

Pqwzz

qwq

t

q

Qwzz

wt

v

v

What happened with the clouds?

Buoyancy is the primary source for the vertical velocity

v

gtw

0

With: )61.01( lvv qq

lvv qq 61.1

K5.0 K3~0 K1~0

Typical numbers: = 0.5K

qv= 1~5 g/kg

ql= 0~3 g/kg

So we need to go back to “down to earth” variables:

{

{ 0),(

,1

0,0

,0

sltsltl

sltl

qqifqqq

a

qqifq

a

c

c

Cloud Scheme in LES: All or Nothing

In Climate models we have partial cloud cover so we need a parameterization.

10Dry Convection

Conditional Instability

•Lift a (un)saturated parcel from a sounding at z0 by dz

•Check on buoyancy with respect to the sounding: vpv

gB

,

0

Stable for unsaturated parcels

Unstable for saturated parcels

Conditionally Unstable!!!

mv

d z

z

dm

v

profile

stableunstable

5.4K/km

CIN

Introductory Concepts 1: Introductory Concepts 1: CAPECAPE

CAPE = Convective Available Potential Energy.

CIN = Convection Inhibition

CAPE

•CAPE and CIN unique properties of moist convection

•Primary Reason why moist convection is so intermittant

CAPE and CIN: An Analogue with ChemistryCAPE and CIN: An Analogue with Chemistry

Free

Energy

Surf Flux

Mixed Layer

CAPE

CIN

Activation (triggering)

LS-forcing

LS-forcing

RAD

LFC LNB

Parcel Height

1) Large Scale Forcing:

• Horizontal Advection

• Vertical Advection (subs)

• Radiation

2) Large Scale Forcing:

slowly builds up CAPE

3) CAPE

•Consumed by moist convection

• Transformed in Kinetic Energy

•Heating due to latent heat release (as measured by the precipitation)

•Fast Process!!

Free after Brian Mapes

Introductory Concepts 3: Introductory Concepts 3: Quasi-EquilibriumQuasi-Equilibrium

0 LSb FJMdt

dCAPE

au

wu

Mb=au wu Amount of convective vertical motion at cloud base (in an ensemble sense)

The convective process that stabilizes environment

LS-Forcing that slowly builds up slowly

Quasi-equilibrium: near-balance is maintained even when F is varying with time, i.e. cloud ensemble follows the Forcing.

Forfilled if :adj << F

Used convection closure (explicit or implicit) JMb ~ CAPE/ adj

adj : hours to a day.

(Arakawa and Schubert JAS 1974)

Introductory Concepts 4: Introductory Concepts 4: Earthly AnalogueEarthly Analogue

•Think of CAPE as the length of the grass

•Forcing as an irrigation system

•Convective clouds as sheep

•Quasi-equilibrium: Sheep eat grass and no matter how quickly it grows, the grass is allways short.

•Precipitation………..

Free after Dave Randall:

16Dry Convection

History of LES of cumulus topped PBL

1. Sommeria, G. (1976) J. Atm Sci. 33, 216-241

2. Sommeria, G and Lemone, M.A (1978) J. Atm Sci. 35, 25-39

3. Beniston, M.G. and Sommeria G (1981) J. Atm Sci. 38, 780-797

4. Bougeault, Ph (1981) J. Atm Sci. 38, 2414-1438

5. Nicholls, L, Lemone, M.A. and Sommeria, G. (1982) QJRMS 108, 167-190

6. Cuijpers J,W,M and Duynkerke, P.G, (1993) J. Atm Sci. 50, 2894-3908

7. Siebesma and Cuijpers J,W,M (1995) J. Atm Sci. 52, 650-666

GCSS; LES intercomparison studies of shallow cumulus:

2006Precipitating trade wind cu

RICO

2000Diurnal Cycle Cumulus

ARM

1998Trade wind cu topped with Scu

ATEX

1997Steady state Trade wind cu

BOMEX

yearCaseExperiment

Siebesma et al. JAS 2003

Stevens et al. JAS 2001

Brown et al. QJRMS 2002

Van Zanten et al. in preperation

0LESscale

large

ttt

0LESscale

large

ttt

BOMEX ship array (1969)

•No observations of turbulent fluxes.

•Use Large Eddy Simulation (LES)

based on observations

•No observations of turbulent fluxes.

•Use Large Eddy Simulation (LES)

based on observations

observed observed

To be modeled by LES

Nitta and Esbensen 1974 JAS

•11 different LES models

•Initial profiles

•Large scale forcings prescribed

•6 hours of simulation

•11 different LES models

•Initial profiles

•Large scale forcings prescribed

•6 hours of simulation

Is LES capable of reproducing the steady state?

Is LES capable of reproducing the steady state?

•Large Scale Forcings•Large Scale Forcings

•Mean profiles after 6 hours•Mean profiles after 6 hours

•Use the last 4 simulation hours for analysis of …….•Use the last 4 simulation hours for analysis of …….

•Turbulent Fluxes of the conserved variables qt and l

•Turbulent Fluxes of the conserved variables qt and l

Cloud layer looks like a enormous entrainment layer!!

lp

l qwc

Lww lvt qwqwqw

LES: “clouds in silico”

mass flux = cloud core fraction * core velocity

Convective Mass flux decreasing with height

xx =

Siebesma et al JAS 2003

Updraft mass flux = updraft fraction * updraft velocity

clouds “in vivo”

Recently validated for “Clouds in vivo” (Zhang, Klein and Kollias 2009)

ARM mm-cloud radar

Conditional Sampling of:

•Total water qt

•Liquid water potential temperature l

Conditional Sampling of:

•Total water qt

•Liquid water potential temperature l

Lateral Mixing between clouds and environment

26Dry Convection

Mass Flux decomposition

u

uuu e

eee

e

e

u

u aa )1(

a

)()1( uu

e

e

u

u wawawaw

M

M-fluxenv. fluxsub-core flux

Courtesy : Martin Kohler (ECMWF)

Siebesma and Cuijpers JAS (1995)

Cloud ensemble:

approximated by

1 effective cloud:

In general: bulk approach:

vvcc

tcc

gBaBwb

z

w

z

M

M

qz

0

22

l

,2

1

1

,for)(

vvcc

tcc

gBaBwb

z

w

z

M

M

qz

0

22

l

,2

1

1

,for)(

M

The old working horse:

Entraining plume model:

Plus boundary conditions

at cloud base.

How to estimate updraft fields and mass

flux? Betts 1974 JAS

Arakawa&Schubert 1974 JAS

Tiedtke 1988 MWR

Gregory & Rowntree 1990 MWR

Kain & Fritsch 1990 JAS

And many more……..

Different tendency to form cumulus anvils is caused by differences in the vertical structure of model mass flux:

M M

, values fixed Mixing; Flexible structure

Tiedtke (1989) in IFS EDMF-DualMSiebesma et al 2007 (JAS)

Neggers et al 2009 (JAS)

•Double counting of processes

•Problems with transitions between different regimes:

dry pbl shallow cu

scu shallow cu

shallow cu deep cu

This unwanted situation has led to:

Swzt

zKw

)( uMw

Standard (schizophrenic) parameterization approach:

Deterministic versus Stochastic Convection (1)Deterministic versus Stochastic Convection (1)

~500 km

•Traditionally convection parameterizations are deterministic:

•Instantaneous large scale Forcing and mean state is taken as input and convective response is deterministic

•One to one correspondency between sub-grid state and resolved state assumed.

•Conceptually assumes that spatial average is a good proxy for the ensemble mean.

Deterministic versus Stochastic Convection (2)Deterministic versus Stochastic Convection (2)

•However: Cloud Resolving Models (CRM’s) indicate that operational resolutions show considerable fluctuations of convective response around the ensemble mean.

•This suggests that a deterministic (micro-canonical) approach might be too restricitive for most operational resolutions.

Mass Flux

pdf

Plant and Craig 2006 JAS

More Sophisticated Parameterization (2)More Sophisticated Parameterization (2)(Plant and Craig 2007)(Plant and Craig 2007)

Parameterization:

•Select N cloud updrafts stochastically according to the pdf

•Calculate impact for each updraft by using a cloud updraft model

•Note : N is a function of the resolution

•Only tested in 1D setting

35Dry Convection

Is this a Cloud??

….and, how to answer this question?

“Shapes, which are not fractal, are the exception. I love Euclidean geometry, but it is quite clear that it does not give a reasonable presentation of the world. Mountains are not cones, clouds are not spheres, trees are not cylinders, neither does lightning travel in a straight line. Almost everything around us is non-Euclidean”.

Fractal Geometry

Benoit Mandelbrot

Instead of

Area-Perimeter analyses of cloud patterns (1)

Procedure:

•Measure the projected cloud area Ap and the perimeter Lp of each cloud

•Define a linear size through

• Perimeter dimension define through:

pAl

llog

pLlog

Slope: Dp= 1

pDp lL

For “ordinary” Euclidean objects:

•Pioneered by Lovejoy (Science 1982)

•Area-perimeter analyses of projected cloud patterns using satellite and radar data

•Suggest a perimeter dimension Dp=4/3 of projected clouds!!!!!

•Confirmed in many other studies since then…

Area-Perimeter analyses of cloud patterns (2)

Instead of Consequences:

•Cloud perimeter is fractal and hence self-similar in a non-trivial way.

•Makes it possible to ascribe a (quantitative) number that characterizes the structure

•Provides a critical test for the realism of the geometrical shape of the LES simulated clouds!!!!

Slope 4/3

Similar analysis with LES clouds

•Measure Surface As and linear size of each cloud

•Plot in a log-log plot

• sDllS )(

•Assuming isotropy, observations would suggest Ds=Dp+1=7/3

31/Vl

Siebesma and Jonker Phys. Rev Letters (2000)

Result of one cloud field

Repeat over 6000 clouds

Some Direct Consequences

Surface area can be written as a function of resolution (measuring stick) l :

3/7,)(2

s

D

L DLlL

lSlS

s

•Euclidian area SL underestimates true cloud surface area S(l=) by a factor

•LES model resolution of l=50m underestimates cloud surface area still by a factor 5!!!

•Does this have consequences for the mixing between clouds and the environment???

1002 sDL

With L=outer scale (i.e. diameter of the cloud) and the normalizing area if measured with L.

2LSL

Transport = Contact area x Flux

x

clKlSlFlSlT

)()()()()( 00000

turbulence diffusive flux

)( 0lSx

clK

)( 0

0l Resolution dependence for transport over cloud boundary (1)

resolved advection

Subgrid diffusion

Consequences for transport over cloud boundary (2)

000000 l

clKlSlFlSlT

)()()()()(

sD

L L

lSlS

2

00 )(

(Richardson Law)

31

00000

/

)()()(

L

lLullullK

!!!!!

D/ s

L

lLSLuclT

370

0 )()()(

No resolution dependancy for Ds=7/3!!

Is this shear luck ????

sD-7/33/4Re

37

)()()(

sD/

LLSLucT

Not really:

Repeat the previous arguments for 0l

Boundary flux T only Reynolds independent if 3/7sD

which completes a heuristic “proof” why clouds are fractal with a surface dimension of 7/3.

47Dry Convection

Gradient Percolation

A stronger underlying mechanism ? (Peters et al JAS 2009)

Dp=4/3

Scale Hierarchy

High Low

low

high

Direct

Numerical

Simulation

Conceptual models

Statistical Mechanics

Self-Organised Criticality

Level of

“understanding”

or

conceptualisation Large

Eddy

Simulations

Global

Climate

SimulationsSimulations

Laboratory

experiments

Atmospheric

Profiling

stations

Field

campaignsSatellite

data

Observations

Models/Parameterizations

resolution

Mixed layer

models

Interface &

microphysical

models