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