Turbulence parametrization

(with a focus on the boundary layer)

Irina Sandu & Richard Forbes

Introduction Irina

Surface layer and surface fluxes Irina

Outer layer Irina

Boundary layer & Cloud evaluation Richard

Why studying the Planetary Boundary Layer (PBL) ?

Natural environment for human activities

Understanding and predicting its structure

Agriculture, aeronautics, telecommunications, Earth energetic budget

Weather forecast, pollutants dispersion, climate prediction

Outline

Definition

Turbulence/Equations

Stability

Classification

Clear convective boundary layers

Cloudy boundary layers (stratocumulus and cumulus)

Summary

PBL: Definitions

The layer where the flow isturbulent.

The fluxes of momentum, heator matter are carried byturbulent motions on a scale ofthe order of the depth of theboundary layer or less.

The surface effects (friction,cooling, heating or moistening)are felt on times scales < 1day.

Free troposphere

50 m

1 - 3 km

Mixed layer

surface layer

Composition

• atmospheric gases (N2, O2, water vapor, …)

• aerosol particles

• clouds (condensed water)

qv

The PBL is the layer close to the surface within which vertical transports by turbulence

play dominant roles in the momentum, heat and moisture budgets.

heat (first principle of thermodynamics)

Governing equations for the mean state

AAA +=Reynolds averaging

gas law (equation of state)

momentum (Navier Stokes)

total water

continuity eq. (conservation of mass)

Simplifications

Boussinesq approximation (density fluctuations non-negligible only in buoyancy terms)

Hydrostatic approximation (balance of pressure gradient and gravity forces)

Incompressibility approximation (changes in density are negligible)

Averaging (overbar) is over grid box, i.e. sub-grid turbulent motion is averaged out.

radiationturbulent

transport

mean

advection

heat

Governing equations for the mean state

AAA +=

total water

Reynolds averaging

gas law p = Rd Tv

virtual temperature

)61.01( lvv qqTT −+=

j

j

j

j

pj

jx

u

x

F

cxu

t

−

−=

+

1

p

v

c

EL

−

Latent heat

release

j

ji

j

i

i

jijci

j

i

j

i

x

uu

x

u

x

Pufg

x

uu

t

u

−

+

−+−=

+

)(12

2

33

mean

advectiongravity

momentum

Coriolis Pressure

gradient

Viscous

stress

Turbulent

transport

precipitation turbulent

transport

mean

advection

j

tjq

j

t

j

t

x

quS

x

qu

t

qt

−=

+

continuity eq. 0=

j

i

x

u

2nd order

Need to be

parameterized !

TKE: a measure of the intensity of turbulent mixing

e

t + u j

e

x j

= g

0

w' v' − ui ' u j'ui

x j

− u j ' e

x j

− 1

ui ' p'

xi

−

turbulent

transport

pressure

transport

buoyancy

production

mechanical

sheardissipation

Turbulent kinetic energy equation

An example :

v’ < 0 , w’ < 0 or v’ > 0 , w’ > 0 w’ v’ > 0 source

v’ < 0 , w’ > 0 or v’ > 0 , w’ < 0 w’ v’ < 0 sink

v

= 1 + 0 .61 qv

− ql

( ) virtual potential

temperature

)(

2

1 222 wvue ++=

PBL: Stability (I)

Traditionally stability is defined using the temperature gradient

v gradient (local definition):

stable layer

unstable layer

neutral layer

How to determine the stability of the PBL taken as a whole ?

In a mixed layer the gradient of temperature is practically zero

Either v or w’ v’ profiles are needed to determine the PBL stability state

0

z

v

0

z

v

0=

z

v

stable unstablez z

v v

PBL: Other ways to determine stability (II)

0vw

Bouyancy flux at the surface:

0vw

unstable PBL

(convective)

stable PBL

0=vw

Or dynamic production of

TKE integrated over the

PBL depth stronger than

thermal production

neutral PBL

Monin-Obukhov length:

s

sv

vwuu

wkg

uL )(,

)(

2

*

3

*=

−=

-105m L –100m unstable PBL

-100m < L < 0 strongly unstable PBL

0 < L < 10 strongly stable PBL

10m L 105m stable PBL

|L| > 105m neutral PBL

PBL: Classification and scaling

Neutral PBL :

turbulence scale l ~ 0.07 H, H being the PBL depth

Quasi-isotropic turbulence

Scaling - adimensional parameters : z0, H, u*

Stable PBL:

l << H (stability embeds turbulent motion)

Turbulence is local (no influence from surface), stronger on horizontal

Scaling : , , H

Unstable (convective) PBL

l ~ H (large eddies)

Turbulence associated mostly to thermal production

Turbulence is non-homogeneous and asymmetric (top-down, bottom-up)

Scaling: H,

sw )( swu )(

3/1

* )(

= Hw

gw sv

v

*

0*

*

0* ,,

w

Q

w

Eq

H

z==

PBL: Diurnal variation

Clear convective PBL

Cloudy convective PBL

ql= qt-qsat

qsat

qsat

Cloud

base

qt lqv l=

PBL: State variables

Clear PBL Cloudy PBL

= Tp

p0

−Rd / cp

Potential

temperature

l -

Lv

cp

q l

Specific

humidityvd

vv

mm

mq

+=

Liquid water

potential

temperature

Total water

content cvd

cvt

mmm

mmq

++

+=

Evaporation

temperature

Conserved variables Conserved variables

Clear Convective PBL

qt qsatv Stephan de Roode, Ph.D thesis

Free atmosphere

Inversion layer

mixed layer

surface layer

Clear Convective PBL

Buoyantly-driven from surface

0.7 H

inversion

level of neutral buoyancy

Main source of TKE

production

Turbulent transport and

pressure term

PBL height:

Entrainment rate: with

(a possible parameterization )

Fluxes at PBL top:

Key parameters: 0)(,,, vve wHw

edtdH ww+=

we =A w*

3

g

0

H v

−=eH ww

3/1

* )(

= Hw

gw sv

v

Clouds effects on climate

Greenhouse effect : warming

Infrared radiation

Solar radiation

Umbrella effect : cooling

Boundary layer clouds

(low clouds)

stratocumulus

High clouds,

like cirrus

Shallow cumulus

Boundary layer clouds over oceans

coldwarmEQ

SST &

surface

windstratocumulus

Shallow cumulus

Sandu, Stevens, Pincus, 2010

Cloudy boundary layers

Stratocumulus

topped PBL

Cumulus PBL

qt qsat v

qtqsa

t

v

Stephan de Roode, Ph.D thesis

Free atmosphere

Inversion layer

mixed layer

surface layer

Free atmosphere

Inversion layer

Conditionally

unstable cloud

layer

surface layer

Subcloud mixed

Dry-adiabatic

lapse rate

wet-adiabatic

lapse rate

Stratocumulus – Why are they important?

Cover in (annual) mean 29% of the planet (Klein and Hartmann, 1993)

Cloud top albedo is 50-80% (in contrast to 7 % at ocean surface).

A 4% increase in global stratocumulus extend would offset 2-3K

global warming from CO2 doubling (Randall et al. 1984).

Coupled models have large biases in stratocumulus extent and SSTs.

Wood, 2012,based on Han

and Waren, 2007Stratocumulus cloud cover (annual mean)

Subsidence

θl (K)rt

(g kg-1)

STBLhSTBL

(m)LWP (kg m-2)

(Liquid Water Path)

rsat(g kg-1)

Characterisation of a stratocumulus topped PBL

0.2g/kg

20g/kg

Subsidence

θl (K)rt

(g kg-1)

hSTBL (m)

STBLLWP (kg m-2)

(Liquid Water Path)

Such a cloudy system is extremely sensitive to thermodynamical conditions

rsat(g kg-1)

Characterisation of a stratocumulus topped PBL

0.2g/kg

20g/kg

Subsidence

θl (K)rt

(g kg-1)

hSTBL (m)

STBLLWP (kg m-2)

(Liquid Water Path)

rsat(g kg-1)

Such a cloudy system is extremely sensitive to thermodynamical conditions

Characterisation of a stratocumulus topped PBL

Subsidence

θl (K)rt

(kg kg-1)

hSTBL (m)

STBLLWP (kg m-2)

(Liquid Water Path)

rsat(kg kg-1)

Characterisation of a stratocumulus topped PBL

Such a cloudy system is extremely sensitive to thermodynamical conditions

0.6K

23/33

l qt

Courtesy of Bjorn Stevens (data from DYCOMS-II)

Warm, dry,subsiding free-troposphere

Entrainment warming, drying

Surface heat and

moisture fluxes

ql

ql,adiabatic

Night-time

Well-mixedThe cloud

deepens

LW radiative

cooling

Processes controlling the evolution of a non-precipitating stratocumulus

24/33

l qt

Courtesy of Bjorn Stevens (data from DYCOMS-II)

Warm, dry, subsiding free-troposphere

Entrainment warming, drying

Surface heat and

moisture fluxes

ql

ql,adiabatic

LW radiative

cooling

Daytime

SW

absorption

The cloud thinsDecoupled

Stabilisation of the

subcloud layer

Processes controlling the evolution of a non-precipitating stratocumulus

Diurnal cycle during observed during FIRE-I experiment

Hignett, 1991 (data from FIRE-I)

Cloud top

Cloud base

8 LT 12 LT 16 LT 20LT 0LT

LWP

Precipitation flux

drizzle

Cumulus

LELE

v

stable

v

unstable

stabledecoupling

Under cloud evaporation affects the dynamics of the boundary layer

Stratocumulus topped boundary layer

Complicated turbulence structure

Cloud top entrainment

condensation

Long-wave cooling

Long-wave warming

Buoyantly driven by radiative cooling

at cloud top

Surface latent and heat flux play an

important role

Cloud top entrainment an order of

magnitude stronger than in clear PBL

Solar radiation transfer essential for

the cloud evolution

Key parameters: 0)(,,, vve wHw

Fzqqw btv ,,,)( 0

Cloudy boundary layers

Stratocumulus

topped PBL

Cumulus PBL

qt qsat v

qtqsa

t

v

Stephan de Roode, Ph.D thesis

Free atmosphere

Inversion layer

mixed layer

surface layer

Free atmosphere

Inversion layer

Conditionally

unstable cloud

layer

surface layer

Subcloud mixed

Dry-adiabatic

lapse rate

wet-adiabatic

lapse rate

Cumulus capped boundary layers

Buoyancy is the main mechanism that forces cloud to rise

Active cloud

CAPE

Cumulus capped boundary layers

Buoyancy is the main mechanism that forces cloud to rise

Represented by mass flux convective

schemes

Decomposition: cloud + environment

Lateral entrainment/detrainment rates

prescribed

Key parameters: 00 )(,)(,, vvb qwwzH

=− wkM duc )(

..

,environ

v

environ

v

z

q

z

PBL: Summary

Characteristics :

several thousands of meters – 2-3 km above the surface

turbulence, mixed layer

convection

Reynolds framework

Classification:

neutral (extremely rare)

stable (nocturnal)

convective (mostly diurnal)

Clear convective

Cloudy (stratocumulus or cumulus)

Importance of boundary layer clouds (Earth radiative budget)

Small liquid water contents, difficult to measure

Bibliography

Deardorff, J.W. (1973) Three-dimensional numerical modeling of the planetary boundary layer, In Workshop on Micrometeorology, D.A. Haugen (Ed.), American Meteorological Society, Boston, 271-311

P. Bougeault, V. Masson, Processus dynamiques aux interfaces sol-atmosphere et ocean-atmosphere, cours ENM

Nieuwstad, F.T.M., Atmospheric boundary-layer processes and influence of inhomogeneos terrain. In Diffusion and transport of pollutants in atmospheric mesoscale flow fields (ed. by Gyr, A. and F.S. Rys), Kluwer Academic Publishers, Dordrecht, The Netherlands, 89-125, 1995.

Stull, R. B. (1988) An introduction to boundary layer meteorology, Kluwer Academic Publisher, Dordrecht, The Netherlands.

S. de Roode (1999), Cloudy Boundary Layers: Observations and Mass-Flux Parameterizations, Ph.D. Thesis

R. Wood (2012), Stratocumulus Clouds, Monthly Weather Review 2012 140:8, 2373-2423

Wyngaard, J.C. (1992) Atmospheric turbulence, Annu. Rev. Fluid Mech. 24, 205-233